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Hahn–Mazurkiewicz Theorem: Suppose $X$ is a nonempty Hausdorff topological space. Then the following are equivalent:

1. there is a surjection $[0,1]\to X$,
2. $X$ is compact, connected, locally connected and second-countable.

It follows that a Hausdorff space satisfying the conditions of (2), then $X$ 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 $n\geq 1$.