I was skipped. There weren't any real problems until we moved to another school district where they never skipped kids. Then some of the teachers became openly hostile to me -- to the point that other students reported it to their parents. I told my mother one of the teachers was picking on me. She didn't believe it until she overheard the other mothers talking about it in the beauty parlor. One teacher told my mother that she felt it was her duty to teach me that math was not fun - in just those words. They took it as a personal affront that I was in their room when I should not be, and did everything they could to fail me. Luckily, it was one of those places where you could pass the course if you could pass the final exam.
You should not expect that all the teachers will love your advanced child, and some might be openly hostile.
You should note that this is not a linear problem. A child who qualifies to be skipped one grade in the first years of school may find themselves three or four years ahead of their peers later on. It is a difference in ability that isn't really solved by giving them a slightly harder set of textbooks.
When I was 12, one of my seventh grade teachers came to me and told me that I had tested equivalent to a college graduate. I could have entered college then. But a 12-year-old in college will have no chance of a social life, and no girl in his classes will give him the time of day. He can't go to bars, or even drive himself around. He is an obvious freak.
On the other hand, if you leave him in the seventh grade, then he can talk to girls his age but you have completely bored him and wasted his time for several years. Moreover, you may have taught him that nothing in life should require any effort, because schoolwork doesn't. He is going to be a freak, anyway. It just won't be so obvious.
Take your pick of problems. I came to the conclusion that the only real way to address it is to teach them to be independent and educate themselves, apart from whatever environment they are in. There really isn't any public program that could properly address those needs, nor could their be, even by skipping grades.
