How do you write the projection operator of a direct sum in terms of basis elements?

Question

Let $U$ and $V$ be two subspaces of $\mathbb{R}^n$ satisfying $U\cap V=\{0\}$, and let $P$ be the projection operator from $U+V$ to $U$. My question is, is there a way to explicitly compute $Px$ for any given $x\in U+V$?

Like if you fix bases for $U$ and $V$, is there a formula for $Px$ in terms of $x$, the basis elements of $U$, and the basis elements of $V$?