Does $\mathbb{R}^n$ is a universal cover for any $n$-dimensional manifold?

Question

I know that any topological manifold has a universal cover (a link: Existence of a universal cover of a manifold.). I was wondering whether $\mathbb{R}^n$ is a universal cover for any $n$-dimensional manifold? I was trying to form a counter example by showing that $\mathbb{R}^2$ doesn't cover $S^2$ but didn't make it.