I would like to give you a different perspective from somebody who has been in a similar situation.
I was your daughter. I was considered very gifted, and generally light years ahead of my classmates. The classic teacher's pet, self driven learner, perfectionist, etc. I qualified for the gifted program immediately, but it was enrichment, not a substitute for curriculum that really was too easy for me. My parents presented it to me as the alternative to skipping grades- I believe it may have been suggested to them. I did not end up skipping in elementary school.
It worked out for me because I went to a small elementary school with teachers that went out of their way to keep me engaged and challenged. I spent a lot of time with the librarian instead of in the classroom, and the even added an honor's reading group so that I would be challenged. It would not have worked for me if I had not had such wonderful teachers.
I tended to have friends who were younger than me (kid who lived in the neighborhood) and was the natural leader. I related better to adults than my age level peers because in maturity and intelligence I was several years ahead. I did have friends my age, but was never considered popular. Basically, I was the Hermione Granger of my school.
A lot changed when I started middle school. My teachers no longer had the option of providing the amount of enrichment I had in elementary school. I still had amazing teachers, such as my eighth grade language arts teacher who allowed me to read my own books instead of the class books I had literally read in the second grade. It wasn't enough, and I ended up skipping the 10-12th grades and started college when I was 15 through the Program for the Exceptionally Gifted (PEG) at Mary Baldwin College.
While we lived in a separate dorm, we had the same academic requirements as the traditional college students, and the majority of my friends were traditional students, not the PEGs who were my same age.
I do not regret that my parents didn't skip me, but I do wonder what would have happened. Would I have felt the need to skip high school? I am technically a drop out- no GED or diploma but I do have a B.A. and soon an MAE. Would I have stood out less, been less of a target for bullies if I were with my intellectual peers? I have no doubt I would have made friends even though it may have been awkward at first.
Please, do not discount the idea just because of social factors. Speaking from experience, the social factors will fall into place easier than you may realize. After a month or so, most kids won't even remember or care that your daughter is younger unless she constantly reminds them. If it had not been for teachers who went above and beyond, I would have been lost to the boredom. Not every school has that luxury, it sounds like yours may not be able to accommodate her the way mine did for me. Please consider doing what is academically appropriate for your daughter. On her current path, it will only get worse as she gets older and continues to outpace her peers.