Often times the reason things don't work out in char 2 is because in such fields $2x = 0$ does not imply $x = 0$. This is why for $o(n,F) \subset sl(n,F)$ you need $char F \ne 2$. The Lie algebra $o(n,F)$ is all $A$ such that $A + A^t = 0$. Taking the trace we see that $0 = tr(A + A^t) = tr(A) + tr(A^t) = 2tr(A)$. If $char F \ne 2$ then we conclude that $tr(A) = 0$ so that $A \in sl(n,F)$.
EDIT: actually for $char \ne 2$ any element in $o(n,F)$ must have zero along the diagonal since $A + A^t = 0$ implies that $2a = 0$ for any diagonal entry $a$. In particular the trace must be zero. But for $char = 2$ a skew-symmetric matrix can have an arbitrary diagonal.
So a similar example of where things get messed up in char 2 is that for char not equal to 2, the space of all matrices is the direct sum of anti-symmetric matrices and symmetric matrices. But in char 2 these spaces are the same since $A = A^t$ if and only if $A = -A^t$ since $1 = -1$.