def.: $\displaystyle SO(3)$ is the collection of lineair orthogonal transformations of $\displaystyle \mathbb{R}^3$

I want to show that two element of $\displaystyle SO(3) $ are conjugated if and only if they have the same trace.

(Maby you want to use: if $\displaystyle A \in SO(3) $ then $\displaystyle \exists\ p \in \mathbb{R}^3, p\neq 0$ and $\displaystyle Ap=p $)