I am currently reading through a book titled "Modern Robotics," and in it, I have encountered a proof of a proposition stating:
Given any $\omega \in \Bbb R^3$ and $R∈SO(3)$, the following always holds: $$ R[\omega]R^T = [R\omega ] $$
Where [ω] is the skew-symmetric matrix representing the axis of rotation, and R is the matrix that details the unit axes of the coordinate frame. The proof is found at this url on page 77-78 (or 95-96 if roman numerals are counted):
http://hades.mech.northwestern.edu/images/7/7f/MR.pdf
A portion of the second part of the proof makes sense to me. Since the cross-product of the rotation axis with each of the coordinate axes will be orthogonal to the axes, the dot product would logically be zero. However, I don't understand why the non-diagonal entries in the first step of the proof are equivalent to the multiplication of an entirely different vector with the rotation axis. I also don't understand how the first part of the proof was derived. So, my first question is why is the cross-product of the rotation axis and coordinate axes contained in each matrix entry in the first part of the proof, and why are these axes multiplied by the vectors that they are multiplied by (so, how is this derived). My next question is: how is the second part of the proof derived from the first? Thanks!