Hahn–Mazurkiewicz Theorem: Suppose $X$ is a nonempty Hausdorff topological space. Then the following are equivalent:
- there is a surjection $[0,1]\to X$,
- $X$ is compact, connected, locally connected and second-countable.
It follows that a Hausdorff space satisfying the conditions of (2) is a quotient of $I = [0,1]$.
Cor: Every connected compact manifold is a quotient of $I$.
Since $I$ is a quotient of $\mathbb{R}^n$, we have your answer.
Cor: Every compact $m$-manifold is a quotient of $\mathbb{R}^n$ for any $n\geq 1$.