  Let $k$ be an algebraically closed field of any characteristic except 2, and
let $G = \GL_n(k)$ be the general linear group, regarded as an algebraic group
over $k$. Using an algebro-geometric argument and Dynkin-Kostant theory for $G$
we begin by obtaining a canonical form for nilpotent $\Ad(G)$-orbits in
$\gl\l_n(k)$ which is symmetric with respect to the non-main diagonal (i.e. it
is fixed by the map $f : (x_{i,j})\mapsto (x_{n+1-j,n+1-i})$), with entries in
$\{0,1\}$. We then show how to modify this form slightly in order to satisfy a
non-degenerate symmetric or skew-symmetric bilinear form, assuming that the
orbit does not vanish in the presence of such a form. Replacing $G$ by any
simple classical algebraic group we thus obtain a unified approach to computing
representatives for nilpotent orbits of all classical Lie algebras. By applying
Springer morphisms, this also yields representatives for the corresponding
unipotent classes in $G$. As a corollary we obtain a complete set of generic
canonical representatives for the unipotent classes in finite general unitary
groups $\GU_n(\F_q)$ for all prime powers $q$.
