Not to necro-thread, but I don't think Ned was a skinchanger/greenseer. We never really hear anything about Ned to suggest that he was.
I am of the opinion that the skinchanging abilities exhibited in Ned and Cat's children actually come from Cat's side of the family. Cat's mother is a Whent, and House Whent is descended from House Lothston. Mad Danelle Lothston was said to have an unnatural control over large bats. So the skin changing could have passed down from House Lothston, to House Whent, to Catelyn Tully, to her children.
Then there's a matter of Jon Snow. Everyone pretty much knows that he's not actually Ned's son, and there is two theories about where he came from. First that he's the son of Rhaegar Targaryen and Lyanna Stark, and second that he is the son of Brandon Stark and Ashara Dayne.
Lyanna Stark was said to be an unnaturally good horse rider, which would make sense if she skinchanged horses.
Brandon Stark was said to have 'The Wolf Blood,' which we do know that the Starks are known for Skinchanging, so it would make sense if he was one too.
Whichever theory you believe would be able to explain Jon.