In the wider setting of simple Lie algebras over $\mathbb{C}$, you are looking for the adjoint orbits of a Lie algebra of type B or D (odd or even orthogonal case).
While this can be viewed concretely as a problem in linear algebra, the more uniform treatment in terms of Jordan-Chevalley decomposition is probably more enlightening. As Robert Bryant comments, there is a big difference between orbits of semisimple matrices (diagonalizable over the algebraically closed field) and orbits of nilpotent matrices. The former orbits are the closed ones, in either the complex or the Zariski topology, while the latter orbits are finite in number but tricky to enumerate.
The subject is classical, so there are quite a few sources available. For a concrete viewpoint, in the language of conjugacy classes in the group (mixed with orbits in the Lie algebra), you might find the old notes by Springer and Steinberg useful, especially Part IV: Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. 131, Springer, 1970. For an approach focused more on Lie algebras and nilpotent orbits, the book Nilpotent Orbits in Semisimple Lie Algebras by Collingwood and McGovern (Van Nostrand Reinhold, 1993) is quite useful. Keep in mind that the Jordan decomposition in your case is essentially the classical one in linear algebra and helps to organize the orbits efficiently. It also works much the same over an algebraically closed field of odd characteristic, though of course characteristic 2 is rather special for orthogonal groups.
