When are $GL_n$ and $GL_m$ equivalent... in characteristic 2?

For fields $K$ and $L$, I am interested in proving that "$GL_n(K)$ and $GL_m(L)$ are isomorphic (as groups) if and only if $m=n$ and $K\simeq L$".

I don't know how generally this is true, but:

• assume $K=L$. In that case, if $char(K) \neq 2$ then there is a usual proof that $n=m$ by looking at the group of involutions of each, that yields $2^m = 2^n$ and this the result.
• if $char(K) = 2$, I do not find any proof of this fact.
• is there a stronger result without supposing $K = L$? Can we at least conclude that $char(K) = char(L)$?

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