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
$$\frac{d\mathbf r}{dt}=\dot{\mathbf{R}}\,\mathbf r_0=\frac{d \mathbf\theta}{dt}\times \,\mathbf r_0$$
Notice
that $~\mathbf\omega = \frac{d \mathbf\theta}{dt}~$ only if the angles are small
edit
with $$\mathbf R(t)=\mathbf R_z(\varphi_1)\,\ \mathbf R_y(\varphi_2)\,\mathbf R_x(\varphi)$$$$\mathbf R(t)=\mathbf R_x(\varphi_1)\,\ \mathbf R_y(\varphi_2)\,\mathbf R_z(\varphi_3)$$
and $$\mathbf \theta=\left[ \begin {array}{c} \varphi _{{1}}\\ \varphi _ {{2}}\\ \varphi _{{3}}\end {array} \right]$$
the angular velocity is:
$$\mathbf \omega= \left[ \begin {array}{ccc} 0&-\sin \left( \varphi _{{1}} \right) & \cos \left( \varphi _{{1}} \right) \cos \left( \varphi _{{2}} \right) \\ 0&\cos \left( \varphi _{{1}} \right) &\sin \left( \varphi _{{1}} \right) \cos \left( \varphi _{{2}} \right) \\ 1&0&-\sin \left( \varphi _{{2}} \right) \end {array} \right] \,\dot{\mathbf{\theta}}$$$$\mathbf \omega= \left[ \begin {array}{ccc} \cos \left( \varphi _{{2}} \right) \cos \left( \varphi _{{3}} \right) &\sin \left( \varphi _{{3}} \right) &0 \\ -\cos \left( \varphi _{{2}} \right) \sin \left( \varphi _{{3}} \right) &\cos \left( \varphi _{{3}} \right) &0 \\ \sin \left( \varphi _{{2}} \right) &0&1 \end {array} \right] \,\dot{\mathbf{\theta}}$$
hence
$$\mathbf\omega\ne\dot{\mathbf{\theta}}$$
for a small angles $~\sin(\varphi_i)=\varphi_i~,\cos(\varphi_i)=1~,\varphi_i\,\dot\varphi_i=0~$
$$\mathbf\omega=\dot{\mathbf{\theta}}$$