Derive Euler angles derivative from angular velocity

I am using a robotics simulator named V-Rep. In V-Rep Euler angles $\alpha$, $\beta$ and $\gamma$ describe a rotation composed by three elemental rotations: $$Q=R_x(\alpha) R_y(\beta) R_z(\gamma)$$

where $R_x$, $R_y$ and $R_z$ represent elemental rotations about axes $x$, $y$ and $z$ respectively of the absolute reference frame. If the angular velocity of an object, whose orientation with respect to the reference frame is described by the Euler angles above, is $\omega=[\omega_x,\omega_y,\omega_z]$ with respect to the reference frame, then am I right to say that $\dot \alpha=\omega_x$, $\dot \beta=\omega_y$ and $\dot \gamma=\omega_z$ ?

What you instead could do is to use quaternion representation of orientation or use a rotation matrix. That would make the maths more easy, the relation between angular velocity and orientation is $\omega = {d\over dt}\operatorname{vec}(q)$, where $\operatorname{vec}(q)$ is the vector part of the quaternion.