Onion,
Your DD sounds exactly like my DD8 when she was at that age. My DD went to public school at 5 and grade skipped to 1st. Even with the grade skip, she got bored after a few months and we had to find ways to challenge her. We did some outside school challenges (kumon for 1.5 yrs) and she was progressing but boredom in school got worse. We had to advocate the school for in-school differentiation.
You should talk to the teachers and ask their recommendations as well as discuss your wish for differentiation. The best scenario would be for her to stay in the grade and do advance work. The grade skipping will work for now but in a few years, you may have to think about another grade skip. And when your DD compete in the talent search or competition, she will be competing against the grade level and not against age and she will have disadvantage.
My DD is happy that we grade skipped her. She wouldn't find friends in the same age class because they would not know when she talk about the books she read and stuff.
Best of luck.
