By coincidence, I put up a protraced working-out of several of these examples today, at http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf

Edit: Over algebraically closed fields (especially in char not $2$, which I want to not think about), these sporadic isogenies are easy to write down. In principle, but I think not in practice for most of us, to see what happens over not algebraically closed fields is a question of "Galois cohomology", as in Weil's "Algebras and classical groups" paper. For me, it's much easier to use a few coordinates. E.g., although $SU(4)\rightarrow SO(6)$ and $SU(2,2)\rightarrow SO(4,2)$, apparently $SU(3,1)$ does not map to $SO(p,q)$ with $p+q=6$. Meanwhile, $SL_2(\mathbb H)\rightarrow SO(5,1)$. Seems weird to me.