We didn't and we are paying the price. My daughter skipped 2nd and was still bored in fourth. By 7th we thought of skipping heragain but decided against it as we were moving to another school district. That was our mistake.
She still finds school boring and calculus too easy as a sophomore. Really the age of a freshman. In retrospect I would have her skip any other younger grade again.
Since she is now in HS I am encouraging her to take as many dual enrollment and AP classes possible but to stay in HS for the 4 years. Cheaper on my pocketbook.
In 7th grade some highly gifted students can sign up for the SAT. Do this. It will open some very good summer school doors for him. Like Great Books Camps and other programs open for the gifted. These are held at some of the best colleges in America and he will attend with other intellectuals. His standardized testing will have to be in the 97%-99% range across the board.
My daughter has had no issues with socialization. She is in Driver's Ed right now with freshman and other sophomores who are older. She started her period early so that was a blessing and she generally hangs better with the more mature kids. Her best friend this year is a senior.
Make sure he does not brag. Have him offer his help to those who are struggling. Be very careful about saying things like You're so smart, intelligent, brainy, etc. Encourage helpfullmess and compassion.
Work on his handwriting. The older kids will have already mastered cursive and typing. Have him practice those this summer.
I would think really hard before skipping fourth. He will do Texas history in depth that year. Maybe you can get some supplements for him. I know my kids loved the state history units. We have four and did VA, NC and CA. (lots of moves, military)