I went to a private school through fourth grade, then we moved to Whidbey and homeschooled for fifth and sixth grade. My mom and I started to fight since we were always around each other. After that, I skipped seventh grade due to high test scores.
I was (and may still be

) a bit socially behind due to the homeschooling and skipping the grade, especially since I didn't have much interaction with other kids during adolescence. As far as being challenged academically by skipping a grade, I wasn't. It's not the grade itself that is a challenge, but the level of your peers. I only had one class in all of middle and high school that truly challenged me. It took going to an Ivy league college to really get pushed.
Besides the social aspect, another bummer about skipping a grade is you are a year younger than other athletes in sports. I still did well in football, wrestling, and track, but I only placed in state twice in wrestling. What really stung is that two guys I had beaten won state in different weight classes the year after I left, and another guy I destroyed took second. My coach still gives my parents a hard time.
Everything has turned out great for me, so I wouldn't go back and change it if I could, but I just thought I would share some of the negatives of skipping a grade. Hope that helps.
As a side note, In my opinion, most public schools suck and are getting worse. For example, many high schools are switching over to a goofy math system that doesn't teach basic skills, but focuses on story problems and crap like that. My brother took through pre-cal in this curriculum, and he sometimes struggles with basic algebra. I call it "fake" math. I definitely benefitted by getting my academic foundation from other sources than public school. If you can't afford private school and aren't able to home-school, then definitely spend a lot of time searching for the best public school you can find.