There's a paper by Burgoyne and Cushman in the Journal of Algebra which answers this for the real field, the complex field and finite fields. My recollection is that there is a paper by Milnor for perfect fields.

There's a paper by Burgoyne and Cushman in the Journal of Algebra which answers this for the real field, the complex field and finite fields. My recollection is that there is a paper by Milnor for perfect fields.

Here are the references:
MR0432778 (55 #5761) Burgoyne, N. ; Cushman, R. Conjugacy classes in linear groups. J. Algebra 44 (1977), no. 2, 339--362.

MR0249519 (40 #2764) Milnor, John . On isometries of inner product spaces. Invent. Math. 8 1969 83--97.