Real projective n-space homeomorphic to sphere mod antipodal points

Question

I want to show that $S^n/C_2$ is homeomorphic to $\mathbb{R}P^n$, where $C_2$ is the group with 2 elements and acts on $S^2$ by $x\mapsto -x$. I think I've found the map $F:\mathbb{R}P^n\to S^n/C_2$ by $[x_0:...:x_n]\mapsto [\frac{x_0}{\sqrt{\sum_0^nx_k^2}},...,\frac{x_n}{\sqrt{\sum_0^nx_k^2}}]$, where in $\mathbb{R}P^2$, bracket notation is used for $x\sim \lambda x$ ($\lambda\neq 0)$ and in $S^n/C_2$, it is used for $x\sim -x$. Firstly, I want to make sure this is indeed a homeomorphism between the two desired spaces, and secondly (most importantly), I don't know how to find the inverse map.