Prove that the fundamental group of $P^2$ the real projective plane is the group with two elements

Question

Presumably the group will just contain $e$ and another $g$. My idea is to use the proof that $\pi_1(S^1)=\mathbb{Z}$ where the covering map from $S^2$ to $P^2$ maps to the line in $\mathbb{R}^3$ through $0$ and $x$. But I only have very vague ideas of how to show this.