starting with
$$\mathbf r=\mathbf R\,\mathbf r_0\tag 1$$
the rotation matrix $~\mathbf R~$ is described with three Euler angles $~\theta_i(t)~$
for a small angles
$$\mathbf R\mapsto I+\left[ \begin {array}{ccc} 0&-\theta_{{3}}&\theta_{{2}} \\ \theta_{{3}}&0&-\theta_{{1}}\\ -\theta_{{2}}&\theta_{{1}}&0\end {array} \right] $$
from here
$$\frac{d\mathbf r}{dt}=\dot{\mathbf{R}}\,\mathbf r_0=\frac{d \mathbf\theta}{dt}\times \,\mathbf r_0$$