There is a $\mathbb{R}^{3}$ with orthonormal basis ($x, y, z$). I want to define other arbitrary orthonormal basis ($a,b,c$), which actually is just rotated ($x,y,z$) basis. The question is can I do it unambiguously by setting dot products $(x,a)$, $(b,y)$ and $(c,z)$? I tried to express $a,b$ and $c$ via dot products or cross products, but got system of trigonometric equations, and I think that there should be more elegant solution.
More generally, what is the best way to define new orthonormal basis by some angles relative to the laboratory system? As far as I understand Euler angles are not defined in case of nutation angle $\phi = 0$ (same as $(c,z)=0$)