Find $\vec{n}$ such that $e^{-i\alpha S_z} e^{-i\beta S_y} e^{-i\gamma S_z} = e^{-i \vec{n} \cdot \vec{S}}$

Question

Any idea how to effciently find vector $\vec{n}$ in the formula $$e^{-i\alpha S_z} e^{-i\beta S_y} e^{-i\gamma S_z} = e^{-i \vec{n} \cdot \vec{S}},$$ where $\alpha,\beta,\gamma$ are given and $\vec{S} = (S_x, S_y, S_z)$ and operators $S_i$ satisfy commutation relations $[S_i, S_j] = \epsilon_{ijk}S_k$? I know how to do it graphically with rotation on a sphere, but I am looking for a different method, solely algebraical, which I could apply to larger groups e.g. SU(3).