The $2\times2$ rotation matrix have the form $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, whose each entry is a weighted sum of 2 (not algebraically independent) parameters $a$ and $b$. How does this property generalize to larger rotation matrices?
Specifically, I want to parameterize each entry in an arbitrary rotation matrix of size $n \times n$ by a weighted sum of arbitrary $m=n$ parameters. What is the minimum $m$ to guarantee to find such a parameterization?