Somehow I got wondering about the following question today:

Suppose $Q_1,\ldots,Q_n$ are random (uniformly sampled) $d \times d$ orthogonal matrices.

What is the expected value of the quantity $\|\sum_i Q_i\|$?

Additionally, suppose I actually generate random skew-symmetric matrices $S_1,\ldots,S_n$, and then obtain corresponding orthogonal matrices via the matrix exponential, $e^{S_i}$.

What is the expected value of the quantity $\|\sum_i e^{S_i}\|$

EDIT: From the comments (and from Mikael's answer) it seems like this is a tough question. But already the case with large $d$ is quite useful.