I was trying to find the angle of an arbitrary rotation matrix, and I decided to use the formula for angle between two vectors:
$\theta=\max\limits_{\vec{x}}\left(\arccos\left(\frac{\vec{x}\cdot R\vec{x}}{\left|\vec{x}\right|\left|R\vec{x}\right|}\right)\right)=\max\limits_{\vec{x}}\left(\arccos\left(\frac{\vec{x}\cdot R\vec{x}}{\left|\vec{x}\right|^2}\right)\right)$.
I think that any higher dimensional rotation lies in a unique plane of rotation, which is the plane (through the origin) such that any vector perpendicular to the plane is rotated by 0 degrees (completely unaffected) and any nonzero vector on the plane is rotated by the largest amount, meaning it is a good choice for $\vec{x}$ in the $\theta$ equation. I saw user856's answer for finding the plane of rotation here, which said I could take two linearly independent columns of $R-I$, and these would span the plane (so any nonzero column of $R-I$ would be a good choice of $\vec{x}$ to find $\theta$).
But the question that that answer was posted under repeatedly said that it was limiting the question to just rotations in a single plane, which seemed to imply that not all rotations in higher dimensions are restricted to a single plane, which might mean that this strategy does not always work. I do not understand higher-dimensional rotations very much, so I am not sure.
My plan will work if every rotation in higher dimensions lies in a unique plane through the origin, where vectors perpendicular to this plane are rotated by 0 degrees, and vectors within this plane are rotated by the largest angle. Is this true? And is there anything else wrong with my method?