Can we always find spin operators?

Question

Fix a positive integer $n$. Can we always find three $n\times n$ complex matrices $S_x,S_y,S_z$ such that

1. all three matrices are self-adjoint;
2. all three matrices have eigenvalues $$\frac{-n+1}{2},\ \frac{-n+1}{2}+1,\ \frac{-n+1}{2}+2,\dots,\ \frac{n-1}{2}-2,\ \frac{n-1}{2}-1,\ \frac{n-1}{2};$$
3. the three matrices satisfy the commutation equations $$[S_x,S_y]=iS_z\\ [S_y,S_z]=iS_x\\ [S_z,S_x]=iS_y?$$

This question arises from trying to find matrix representations of the spin operators describing a spin-$\frac{n-1}{2}$ particle. I believe it has something to do with representation of $\text{SO}(3)$ but please assume I don't know anything as I was not taught representation theory in quantum mechanics classes.

(If $n=1$, all three matrices are just zero. If $n=2$, we could use half of Pauli matrices.)