My daughter's best friend in 1st grade, we'll call her Kate, skipped second grade for the first 6 weeks of that school year because her parents went to all the trouble with the school and the school district to get her to skip. Kate too was involved in the school's talented and gifted prorgram, she has a birthday in October so she was close in age to the older class and in first grade, the teacher had an advanced program in class for students who were not challenged by the regular assignments for students who were more "average" in academics and Kate participated in that program as well. Kate's mom and dad had to meet with the adminstrators several times, the counselors and eventually, Kate had to pass placement tests before they would let her skip a grade. Her parents bought all the 2nd grade books and tutored her all thru the summer and before 2nd grade began, she took the placement tests and placed out of 2nd grade. She skipped 2nd grade and started in 3rd grade on the first day of school. Unfortunately, after all that hard work by her and her parents, Kate was miserable in third grade.
Her mom once mentioned to me back when Kate and my daughter were in first grade that she didn't like Kate's first grade teacher because she had made a comment along the lines of "your daughter is right on schedule behaviorally, socially and emotionaly with a first grader." Her mother, for some reason, took insult to that, but that ended up being the demise to their plan to have Kate advance. She missed all her friends and felt alienated from the "older" kids who had gone thru second grade already and were in third grade with her. After 6 weeks, her parents gave her the option to move back in to second grade and she took it. She's been a very content happy talented and gifted child ever since.
So my advice to you is this. Make sure your daughter is also as advanced behaviorally and socially as she is academically before you try to move her up or all your work may be for nothing. Kate's mother said she wishes they would never tried having her skip because it was a lot of work and effort and she wouldn't recommend it to anyone. It doesn't seem that public schools are as conducive to moving kids up as private schools so if you're really looking to have your daughter skip a grade, you might want to consider putting her in a private school where that is more commonplace.
Definitely check to see if your school offers the advanced work option that I described in my note about first grade. My daughter was also part of that group in first grade and she got a lot from doing the advanced work in class. That may just be something specific to her school or her school district, but it was great and I think it really helped her to stay interested in school because it was definitely more challenging that way.
Good luck in whatever you decide. :-)