Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller?
In $\mathbb{R}^2$ the rotation is characterized by a single angle. This angle can be interpreted as the magnitude of the rotation.
In $\mathbb{R}^3$ however we have three possible axes around which to rotate. Furthermore we can get a zero rotation by rotating around each axis by $\pi$ radians. Therefore devices such as the sum of the three angles do not make sense.
In $\mathbb{R}^n$ the situation is even more complicated.
Let's say that the rotation is characterized by matrix $R$. Is the distance of $R$ from the identity matrix $I$ of the appropriate dimension a reasonable choice? For example $$m_{R} = || R - I|| $$ where $||.||$ is some matrix norm. If so, which norm would be appropriate?