Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed the integral over the orthogonal group, $$ \int_{O(N)} \prod_{k=1}^{2n}u_{i_kj_k}du=\sum_{\sigma,\tau}\Delta_\sigma(i)\Delta_\tau(j) {\rm Wg}_N(\sigma^{-1}\tau),\qquad (1)$$ where the sum is over matchings, $\Delta_\sigma(i)=1$ if and only if the sequence $i$ satisfies the matching $\sigma$ and ${\rm Wg}_N$ is called the Weingarten function.

This implies for instance that $\int_{O(N)} u_{11}u_{22}du=0$ because the list $(1,2)$ does not match.

On the other hand, we know that a matrix from $SO(2)$ is of the form $u=\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{pmatrix}$ so clearly we have $\int_{SO(2)} u_{11}u_{22}du=1/2$. This shows that the $SO(N)$ result can be quite different from the $O(N)$ one.

Is there a general theory of integrals like (1) over $SO(N)$?