How to write explicity a curve on $S^n$?

Question

I considered the $n$-sphere $S^n=\{x\in \mathbb{R}^{n+1}| \space ||x||=1 \}$ and $p\in S^n$.

I want to write down explicity a curve $\sigma$ on $S^n$ passing through $p$ (for example one of the great circle passing through $p=(p_1,...,p_{n+1})$).

How can I write $\sigma$ ? For $S^2$ I would use the spherical coordinates and find the parametric equation. Is there something analogous for $S^n$?

Thanks for the help!