One other thought: you say that she has poor social skills. Is that with kids her own age, or with ALL people?
Many HG+ kids do poorly socially with age-mates. It's not uncommon. Sometimes their social skills improve dramatically when they are placed with kids the same intellectual age as they are. That's not a blanket true statement, but it is something to consider, I think.
Have you tried her with older kids academically or socially? How does she respond? Does she have any friends older than she is?
I'd start there if you haven't already. You might find that her immaturities are really manifestations of her frustrations with people who "don't speak her language," and that she behaves like a much older child around older children.
If not, then a grade skip is definitely a bad idea, I'd say. But I wouldn't necessarily rule out the skip until you try her with older kids.
FWIW...
