 Solution

Choose the unit vector $\mathbf{p}=(1,-1,0)/\sqrt{2}$, which is perpendicular to $\mathbf{v}$. Then $M\mathbf{p}=(0,1,-1)/\sqrt{2}$. Their dot product is $-1/2$, and the rotation matrix preserves their norms, soThe same unsigned angle holds for every nonzero vector perpendicular to the axis: this is the rotation plane of a three-dimensional proper rotation. Alternatively, $\mathrm{tr}M=0=1+2\cos \chi$ gives the same rotation angle $\chi =2\pi /3$.