What is the Isomorphism between the adjoint representation of SU(2) and its representation on rank 2 symmetric tensors? Let su(2) be the Lie algebra of SU(2), thought of as a representation of SU(2) by conjugation. 
Let S be the rank-2 symmetric tensors over the complex numbers, acted on by SU(2) in the obvious way.
I have seen it asserted that the complexification of su(2) is isomorphic (as a representation) to S, and presumably there is some simple explicit isomorphism that should be obvious.  Alas, to me it is not obvious.  What is that isomorphism?

The map $C^2 \otimes C^2 \ni v \otimes w \mapsto v w^t$ allows us to identify the space of rank-2 symmetric tensors with the space of symmetric $2\times 2$ matrices, which I'll call $Sym$.  The corresponding action of an element $A \in SU(2)$ on such a matrix $B$ is then $ABA^t$.  The complexification of $su(2)$ is $sl(2,C)$, the space of traceless $2\times 2$ complex matrices.  An isomorphism of these two reps is given by the map
$$
sl(2,C) \to Sym, B \mapsto B \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right).
$$
That this is a map of representations follows from the fact that
$$
A^{-1} \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right) = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right) A^t 
$$
for $A \in SU(2)$.  This fact is easily verified by using the fact that every element in $SU(2)$ is of the form
$$
\left(\begin{array}{cc} z & w \\ -\bar w & \bar z \end{array}\right)
$$
with $|z|^2 + |w|^2 = 1$.
I found this isomorphism by just playing around, and I think I just got lucky.  There is probably another one that is less ad-hoc.
