From Hilbert 90 (or, more precisely, a generalisation thereof), we know that the first (Galois) cohomology group of $GL_n$ is trivial, no matter the field of definition. However, for unitary groups, which are outer forms of the general linear group, the story is different. As I have heard again and again, the first cohomology group is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. Is there a quick proof? I suspect (based on almost nothing) that this has to do with the fact that a unitary group splits over a quadratic extension. Hence, the question is : could someone give a proof/reference of this fact?
Added: What I would be looking for is a way to explicitly describe the non-trivial cocycle in $H^1(F,U(p,q))$, if this is at all possible. In fact, to be more precise, I am looking at $U(n,n)$ over $F$, where $F$ is either a number field or a non-archimedean local field.