Velocity of a point expressed by using rotation matrix between two reference frames

I'm trying to express the velocity of a point P by using a rotation matrix between two reference frames (see attached file please).

Notation: the vertical bar with $$O_1$$ or $$O_2$$ means that the vector's components are expressed in the $$O_1$$ or $$O_2$$ reference frame. The dot over a vector means derivative with respect to time of that vector.

Is my procedure correct? In the last equation I used the Poisson's formulas to express the derivative of the rotation matrix.

Thank you so much.