Video was posted on the internet of rebels defacing a poster of President Assad at the Bab al-Hawa crossing
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Officials killed: As it happened
City 'in grip of fear'
Time to intervene?
Profiles of officials killed
Syrian rebels have captured a number of positions on the country's borders with Turkey and Iraq.
A senior Iraqi official said all the crossings on Syria's eastern frontier had been seized. At one point, two Turkish posts were also in rebel hands.
The push came a day after a bomb claimed the lives of three senior defence officials in Damascus.
The first images of President Assad since the attack have appeared, largely ending rumours he might have been hurt.
The footage appeared to show Gen Fahd Jassim al-Furayj, chief of staff of the armed forces, being sworn into his new post.Escalating battle
The rebels, perhaps sensing that the regime was too preoccupied with the escalating battle for the capital, stormed all the posts on the Iraqi border, the BBC's Jim Muir in neighbouring Lebanon says.
The major Abu Kamal crossing on the Euphrates river in the east was captured after a clash with government forces, opposition activists said.
More than 20 Syrian soldiers and their commander were killed when a remote army outpost in the far north-east was attacked, Associated Press news agency reported.
Iraq's government, seen as sympathetic to President Bashar al-Assad, has threatened to shut its side of the border and one official told Reuters news agency that it was closing the Abu Kamal crossing.
On the frontier with Turkey, too, rebels were said to have taken control of two posts, at Bab al-Hawa and Jarablus.
Video from the Bab al-Hawa crossing in Idlib province soon emerged of rebels defacing a portrait of President Assad, but they later reportedly withdrew from the position.
For four days, rebels have been involved in clashes in areas of the capital as they push their "Damascus volcano" operation against Syrian armed forces.
Damascus-based activist Hassan describes how people are too afraid to venture outside
The deaths of three top security officials has led to a mobilisation of government troops in an attempt to drive the rebels out of the city.
The president's brother-in-law, the defence minister and head of the government's crisis team were killed by a bomb as they attended a meeting at the national security headquarters.
Tanks and armoured vehicles were reported to have moved into Qaboun on Thursday, close to the centre of Damascus.
There were heavy casualties, activists said, as a result of an army bombardment of Zamalka in the eastern outskirts of Damascus.
The London-based Syrian Observatory for Human Rights put the number of fatalities across the country on Thursday at 250.'Indefensible'
The pace of events in Syria was in marked contrast to the diplomatic stalemate at the UN Security Council, where Russia and China vetoed a Western resolution calling for tougher sanctions on Damascus.
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Laura TrevelyanBBC News, New York
The mood inside the Security Council chamber was acrimonious after China and Russia vetoed the resolution. Britain's ambassador accused the two nations of protecting a brutal regime by their actions. America's ambassador said the council had failed utterly in the most important task on its agenda.
China's ambassador denounced what he called an uneven resolution which placed pressure on one side, while Russia's representative claimed the resolution would have opened the path to military involvement in Syria's affairs.
Now negotiations are under way to try to extend the mandate of the UN monitoring mission in Syria which is due to expire on Friday.
The mission is supposed to monitor a ceasefire and support a political process - neither of which exist. So the UK is proposing a 30 day "final" extension.
Under the Western-backed plan, the Damascus government would have been threatened with non-military sanctions under Chapter Seven of the UN Charter if it failed to move troops and heavy weapons from populated areas.
But the use of Chapter Seven paved the way for "external military involvement in Syrian domestic affairs", Russia's UN ambassador Vitaly Churkin argued.
The UK, US and France said the UN had failed the people of Syria and UK Foreign Secretary William Hague condemned the use of the veto as "inexcusable and indefensible".
The Security Council still has to decide whether to renew the mandate of the 300-strong UN observer mission in Syria, due to end on Friday, and talks on a possible compromise resolution have begun.
The UK is said to be proposing an extension for a "final 30 days".
Although the US says it might consider a brief addition to the monitoring mission, White House spokesman Jay Carney said Washington did not back sending in unarmed UN employees "when there's no mechanism within the resolution to create consequences for the regime".