Let $Q \in \operatorname{Conv} SO(3)$.
Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}}, R_{i} \in SO(3)$?
An approximation can also be useful.
A closely related answer: The convex hull of SO(n) is studied in this paper by Saunderson, Parrilo, and Willsky (2015). In particular, that paper proves that this convex hull is doubly spectrahedral, and provides explicit representations. For $n=3$, see (1.11) in the paper.