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) 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 $n\geq 1$.