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
$$\mathbf R(t)=\mathbf R_x(\varphi_1)\,\
\mathbf R_y(\varphi_2)\,\mathbf R_z(\varphi_3)$$

and 
$$\mathbf  \theta=\left[ \begin {array}{c} \varphi _{{1}}\\  \varphi _
{{2}}\\  \varphi _{{3}}\end {array} \right]$$

the angular velocity is:

$$\mathbf \omega= 
 \left[ \begin {array}{ccc} \cos \left( \varphi _{{2}} \right) \cos
 \left( \varphi _{{3}} \right) &\sin \left( \varphi _{{3}} \right) &0
\\  -\cos \left( \varphi _{{2}} \right) \sin \left( 
\varphi _{{3}} \right) &\cos \left( \varphi _{{3}} \right) &0
\\  \sin \left( \varphi _{{2}} \right) &0&1
\end {array} \right] 
\,\dot{\mathbf{\theta}}$$

hence

$$\mathbf\omega\ne\dot{\mathbf{\theta}}$$

for a small angles 
$~\sin(\varphi_i)=\varphi_i~,\cos(\varphi_i)=1~,\varphi_i\,\dot\varphi_i=0~$ 

$$\mathbf\omega=\dot{\mathbf{\theta}}$$