Moments of fixed observable in the eigenbasis of a Gaussian Orthogonal Ensemble

Question

I need a clear derivation of the moments of a fixed observable $\hat{O}$ in the eigenbasis of a Gaussian Orthogonal Ensemble or a Gaussian Unitary Ensemble. Write $\hat{O} = \sum_i o_i |o_i><o_i|$. $U_{\alpha i}$ denotes a transformation function between bases $|E_{\alpha}>$ and $|o_i>$, where $|E_{\alpha}>$ is the eigenbasis of the Hamiltonian. If $U$ is distributed uniformly with respect to the unitary Haar measure then I know that I should have $<U_{\alpha i}U_{i \beta}> = \frac{\delta{\alpha \beta}}{d}$ where $d$ is the dimension of the Hamiltonian. How do I obtain this and other moments in a clear way?