Rotationally-uniform sampling over an $\rm SO(3)$ representation

Say that I have a representation $V_\ell$ of $\rm SO(3)$ , with angular momentum number $\ell$ . In Dirac notation, states in $V_\ell$ can be represented in the form $|v⟩ = \sum_{m=-\ell}^\ell c_m |\ell,m⟩,$ where, if the representation is real-valued, we require $c_{-m} = (-1)^m c_m^*$ . I would like to randomly sample from this space , i.e., the vector $|v⟩$ and/or its coefficients $c_m$ , in a way which is invariant under the action of $\rm SO(3)$ on this representation. I'm hoping this is a standard result -- I'm grateful for any pointers, or, more generally, for explicit answer =).