Since nobody has attempted to say what happens in even characteristic, let me do that briefly.

As someone observed earlier in a comment that seems to have been deleted, in even characteristic and even dimension, alternating bilinear forms are symmetric. There is a unique isometry class of such forms - the matrix of one has 1's on the "antidiagonal" and 0's elsewhere. The group that preserves such a form is the symplectic group ${\rm Sp}(2n,2^e)$.

This group, however, has two subgroups, which preserve the two types of quadratic form in characteristic 2 (the convenient one-one correspondence between symmetric bilinear and quadratic forms does not work in characteristic 2), and these are the orthogonal groups ${\rm GO}^+(2n,2^e)$ and ${\rm GO}^-(2n,2^e)$. All of their elements have determinant 1, so they are equal to ${\rm SO}^+(2n,2^e)$ and ${\rm SO}^-(2n,2^e)$.

These groups are not perfect and have subgroups of index 2, often denoted by ${\Omega}^+(2n,2^e)$ and ${\Omega}^-(2n,2^e)$, which consist of those elements that are a product of an even number of reflections. (The homormorphism from the special orthogonal group to the cyclic group of order 2 is still usually called the spinor norm homomorphism, although its definition is not identical to the one in odd characteristic.) These have trivial centres (in dimension at least 4) and so are isomorphic to their projective versions, and the $\Omega$ groups are simple in dimension at least 6.