3-D Golden Angle

Question

I'm trying to find a 3D analog for the golden angle, g. In particular, I'm interested in the following property of the golden angle: Let $M_2$ be the rotation matrix that rotates by g. Then, for all n>1, $$ n^2\left | I_2-M_2^n \right |>\gamma_2 $$ For some constant $\gamma_2>0$, where $I_2$ is the 2x2 identity matrix. For the golden angle, $\gamma_2=\frac{4\pi^2}{5}$. I think this uniquely defines g, if we add the condition that $|M_2|=1$, up to a sign (not sure how to prove that). By analogy, I'm wondering how to solve for the analog in higher dimensions. Namely, I'm wondering if there is a 3x3 matrix $M_3$ and a constant $\gamma_3>0$ such that, $|M_3|=1$, and for all n>1 $$ n^3\left | I_3-M_3^n \right |>\gamma_3 $$ I'd greatly appreciate any and all help or any suggestions.