  In motion Kinematics, it is well-known that the time derivative of a
3x3rotation matrix equals a skew-symmetric matrix multiplied by the rotation
matrix where the skew symmetric matrix is a linear (matrix valued) function of
the angular velocity and the rotation matrix represents the rotating motion of
a frame with respect to a reference frame. The equation is widely used in
engineering, e.g., robotics, control, air/spacecraft modeling, etc. However,
the derivations found in the literature are indirect. Motivated by the fact
that the set of 3x3rotation matrices, i.e., SO(3), is a Lie group, forming a
smooth (differentiable) manifold, we describe the infinitesimal increment of
the rotation matrix in terms of rotation matrices and show that the above
equation immediately follows.
