It's always good to question things like this- propaganda only hurts animals in the end.
I think the main point behind the 'animals are skinned alive for their fur' activist soundbite is that a lot of fur comes from China where there are no animal welfare laws and they can (and do) do what they want. With profit and improved 'productivity' also comes immense suffering for convenience.
While footage has captured animals being skinned alive (because the warmth of their bodies makes the process cleaner), it is also the norm to be vaginally/anally electrocuted. After all, they cannot get "mess" on their precious fur because they have to sell it! It's standards like this which make the anti-fur argument stand against any opponent (any opponent with a conscience, that is).
I'll be honest, I won't watch anything to do with fur because the few things I have glimpsed have scarred me for life, but you might be interested to watch a new documentary, "Skin Trade"
"You wonít find any urban legends or conspiracy theories here; the fur trade is real, atrocious, and outdated...One word of warning: This film isnít for the faint of heart. Raw, brutal footage from fur farms in the United States and abroad shows animals being maimed and tortured in traps, as well as subjected to electrocution and fatal beatings".
It sounds to me like this film will be the truest presentation of what really goes on. I know you can view it online (but there's only a trailer in this link).