  Let $W$ be a random positive definite symmetric matrix distributed according
to a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inverse
matrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ...
W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingarten
function, which was recently introduced in the study for Haar-distributed
orthogonal matrices. As applications, we give formulas for moments of traces of
a Wishart matrix and its inverse.
