Let $V$ be oriented two-dimensional Euclidean space. Then we can define an oriented angle $\phi$ between two nonzero vectors $u,v\in X$ by formulas: $ \phi=\arccos \frac{\langle u,v\rangle}{\|u\| \|v\|}$ if $(u,v)$ is a positive basis or $u,v$ are linearly dependent, and $\phi=2\pi-\arccos \frac{\langle u,v\rangle}{\|u\|\|v\|}$ if $(u,v)$ is a negative basis in $X$..
Let $X$ be a three dimensional Euclidean space. Let's consider a rotation different from identity $f\in SO(3)$. Then $f$ has a two dimensional invariant subspace $V$. Then its orthogonal complement $V'$ is a one dimensional invariant subspace. Let $V'$ be also oriented. Then orientation on $X$ and $V'$ induce an orientation on $V$: we fix a positive basis $e_1,e_2,e_3$ in $X$ and a positive basis $h$ in $V'$ and we call a basis $f_1,f_2$ in $V$ a positive iff $f_1,f_2,h$ is positive in $X$.
How to determined oriented angle of rotation, that is oriente angle between $u$ and $f(u)$, for $u\in X$ ?
Thanks
P.S. I know how to determine nonoriented angle $\psi$: $Tr f=2\cos \psi+1$.