Is there such thing as an $n$-fold cover of $SO(n)$?

Question

I know already that the $Spin(n)$ group is a double cover of $SO(n)$, but is there such a thing as a triple cover, fourfold or $n$-fold cover?

I am interested to know if there are 3-spinors (?) or the likes of them, or is there some simple reason why such things are not possible.

Answer 1

No. The fundamental group of $SO(n)$ is $\Bbb Z/2$, so its universal cover is just a double cover. Indeed, what this is saying is that $Spin(n)$ is simply connected.

Answer 2

If $n=2$ $SO(2)\cong S^1$, so its fundamental group is $\mathbb{Z}$ and the universal cover is $\mathbb{R}$; if $n\ge 3$ then $Spin(3)$ is simply connected and a double cover of $SO(3)$; so you cannot find a $n$-fold cover for $n\ge 3$.