The structure of classical groups goes back a long way and has been treated in a number of books, but in varying generality (arbitrary fields, various commutative rings, etc.).   One older source in French is J.A. Dieudonne's concise Springer Ergebnisse volume *La geometrie des groupes classiques* (1963).   A probably more readable modern textbook with limited aims is Larry Grove's *Classical Groups and Geometric Algebra* (AMS, 2002), just cited by Skip.   There is also Emil Artin's old book *Geometric Algebra* and a much larger book by Hahn-O'Meara oriented more to algebraic K-theory.   Anyway your group is simple both as an algebraic and as an abstract group (special orthogonal groups in odd dimension are also adjoint groups).   There's no real need to get into algebraic groups, BN-pairs, or the like, though this is the "correct" general setting as Tits showed.

Actually, simplicity of various classical groups is proved sometimes in graduate algebra textbooks (which I don't have at hand).    It depends how far you want to go.  Over more general fields, especially of characteristic 2, a little more care is needed but these groups are still simple or very close to it even over most finite fields.