I had a similar problem when my son was that age! We did have him tested and found out that not only was he gifted (and bored!), he also has ADHD! Now the ADHD is treated and he is in a gifted program and doing very well most of the time. He is 13; finishing 7th grade. He still continues to challenge adults, because he is very intelligent and he knows it! This has gotten him into some situations and he's learning when to just be quiet and listen to what he's told to do....but it's been a gradual learning process.
We knew he was smart even in preschool; he could already read way ahead of his level but we didn't have him skip any grades because of his emotional/social immaturity. Looking back, that was a good decision! School is about learning but also about learning to interact with others and deal with social situations. I think in this area he has been the most challenged. Most of his teachers were willing to work with him individually until third grade...then he had a teacher who just expected him to sit still and do his work because at that age most of the other kids were able to do that! This is when we diagnosed the ADHD and treating it really helped. I know alot of people are reluctant to medicate their kids but even he notices the difference when he forgets to take his pill! The medication helped control his impulsiveness and distractability so that he could at least follow along in class (not be daydreaming or reading something else!). We still needed to understand that part of this was just his personality and we had to adjust, too. For example, his piano teacher actually quit because she would tell him how to play a song, then he would decide he's rather play it a different way because his way sounds better! We had the talk about respecting what your teachers tell you even if you think something else.....but eventually we gave up trying to force him into a mold when it came to extracurricular activities. It is enough to ask him to hold it together during school hours and then allow him lots of free time to be creative and explore on his own time.....within safe limits of course!
A consolation to both you and I is that while these kids are challenging to raise, they grow into the most successful adults! The key is to choose your battles and focus on the most important things. And having him tested is a great idea even if you don't want to treat with medication...at least you will know what you are dealing with and can educate yourself on how to best handle the situation. I can recommend a few good books if you write back to me. Hope my experience has helped you!!