$A_5$ as finite subgroup of $SO(5)$, trigonometric formulas for rotation matrix, and being a simple group

Question

Over the reals, $SO(2)$ and $SO(3)$ have well known trigonometric expressions as "physical" rotation matrices. I think also $SO(n)$ can be described by trigonometric functions by introducing additional angles. The alternating group $A_n$ is a finite subgroup of $SO(n)$. When going from $A_4$ to $A_5$, the alternating group becomes simple (actually simple for $n \neq 4$). Is this somehow reflected in some kind of structural change in the corresponding trigonometric representation of $SO(n)$ for $n \geqslant5$?