I agree with the other posters about evaluating your daughters social maturity. I was a late November baby and when I started school, the cut off date for Kindergarten was 12/31 so I was sent to school. I was academically and socially more than ready. My problems started developing during 3rd Grade and continued through high school. I was small for my age, though mature, and I had learning differences that weren't even identified in those days. I graduated from High School at 17 1/2 and left home for college a few months later. I was too young.
I have several friends whose children have skipped a grade in school, and now they are all graduating at 16 and 17 years old, and in my opinion much too young to be heading off to college and the influences of the older students. One friend mentioned that she wished that she had never let her son skip a grade. He is and always has been mauture for his age, but as she said, she is losing him a year sooner. Another friends daughter was the youngest in her class to begin with, and then skipped a grade. She will be 16 and 9 months when she graduates. As a sophomore in HS and 14, she is already dating, (even went to PROM) and has a boyfriend. On the opposite end of the spectrum, my daughter who is her friend repeated a grade and will be entering HS at having just turned 15. The friend is more mature in some areas, but when they are together, they are still little girls. This girl will barely even have her drivers license when she graduates. I can't imagine sending my child off to college with such little driving experience.
Ask the school if they can provide a more challenging curriculum, and no, that doesn't mean more work or homework. If this is something they can do, and even allow her to attend some classes with the next grade, I would seriously consider going that route.
Have you considered homeschooling? The K-12 Curriculum is way advanced over normal public schools. They test the child and adjust the curriculum accordingly. Your child could be enrolled in 3rd Grade, but taking 5th Grade reading and math, and working on 2nd Science. The K-12 curriculum is an online program through virtual schools. Much of the work in the lower grades is done in workbooks and on paper, but many of the classes or portions of the classes are done online.
Think about and look into the future. Do you want your child heading off to college at 17 with the influences, both good and bad, that will abound in college? Do you want her being the one in her group of friends staying home on a Friday night because you have set the boundry of no dating until 16, and her friends are all out on dates? Would she possibly be the Valedictorian or Salutitorian if she remains with her class but miss out on that opportunity by being the youngest in her class? Will you be financially prepared to send her off to college a year early?
I also agree with Margie. You need to be aware of the curriculum content, not just for 3rd Grade, but even down the road a few years. Much of the reading is "required", how will you handle that? As mature as my daughter was in 5th Grade, she wasn't mature enough to read "The Diary of Anne Frank" at that age which was required reading in 6th Grade. She also read "The Prince and The Pauper" in 6th Grade, which is a very difficult book for most adults.
There are many things for you to cinsider, and I wish you the best of luck. Personally, I would keep my daughter in her current grade and ask for a more challenging curriculum. Many times teachers want to skip the child a grade because it requires less work for them. (More planning, lesson plans, grading a different set of papers etc.) Many times I have seen parents push to get their child moved up a grade, and it isn't always in the best interest of the child, but an ego trip for the parents.