For any $m<n$ the $n\times n$ unitary matrix $\Omega$ has the block decomposition
$$\Omega=\begin{pmatrix} A&B\\ C&D\end{pmatrix},$$
where $A$ has dimensions $m\times m$, $D$ has dimensions $(n-m)\times(n-m)$, $B$ has dimensions $m\times(n-m)$ and $C$ has dimensions $(n-m)\times m$. Up to a set of measure zero, the matrix $D$ will not have a unit eigenvalue, so $I-D$ is invertible. We then define the continuous map $F$ from $U(n)$ to $U(m)$ by
$$F(\Omega)=A+B(I-D)^{-1}C.$$
One readily checks that $F(\Omega)$ is unitary$^\ast$ and as David Speyer points out $F(\Omega)$ inherits$^{\ast\ast}$ the Haar measure from $\Omega$.

_{
$^\ast$ More generally, for any $\Omega\in U(n)$ and $V\in U(n-m)$ the matrix $U=A+B(I-VD)^{-1}VC$ is unitary. One can think of $V$ as the reflection matrix of a barrier that closes off $n-m$ scattering channels. Then the remaining $m$ channels have scattering matrix $U=A+\sum_{k=0}^\infty B(VD)^kVC$, where $k+1$ counts the reflections off the barrier. As a further check for the unitarity of $U$, I can offer a Mathematica Notebook.
$^{\ast\ast}$ Since for any $g\in U(m)$, $G={{g\;\;0}\choose{0\;\;I}}\in U(n)$ one has $F(G\Omega)=gF(\Omega )$, the measure remains left-invariant.
}