Washington correspondent Gary O'Donoghue, meanwhile, said the incident was "an incredibly violent attack", and blamed Trump's rallies for pitting the supporters of the president against the media.
However, BBC Washington Editor Eleanor Montague, whom he was with, told BBC the protester had attacked other crews within the rally but Skean "got the brunt of it". "Fortunately our cameraman is fine, he is made of stern stuff", he told the Today programme.
Footage captured from Mr Skeans's camera was posted to Twitter by Mr O'Donoghue and shows the shot focused on Mr Trump before jolting away.
The footage then shows the man who allegedly pushed Skeans being dragged away by security as he shouted, "Fuck the media".
He is being restrained by a man in a black t-shirt - also wearing a red hat - as he shouts "f*** the media, f*** the media".
According to a BBC statement, this time Trump "could see the incident and checked with us that all was OK".
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Prior to the attack in El Paso, where Trump had chose to rally support for his contentious border wall, the president had regularly referred to the media as "fake news".
'It is clearly unacceptable for any of our staff to be attacked for doing their job'.
Trump's rallies have for a long time featured the president turning his supporters against the media, whom he accuses of being "fake news".
'I have been spat at before, they hurl abuse at American colleagues in particular'.
Trump told the crowds inside El Paso County Coliseum that 69,000 people had signed up to attend.
Instead, he mocked Mr O'Rourke, insisting the Texan has "very little going for himself except he's got a great first name" and deriding his crowd size, even though both men drew thousands. 'It's a moral thing, to put it in the opposite terms. So we are saying finish the wall.
Donald Trump was speaking at a rally in El Paso, Texas.