Finding the best approximation of orthonormal vectors has the largest square inner product with a set of given vectors

Question

Suppose we are given a set of fixed vectors $v_1,\dots,v_m\in\mathbb{R}^n$ and assuming $m\leq n$. I want to find an orthonormal set $[u_1,\dots,u_m]$ such that the next quantity is maximized $$ \max_{U\in \mathbb{R}^{n\times m}~~~:~U^\top U=I_m} \sum_{i=1}^m|\langle v_i,u_i \rangle|^2 ,\quad \text{where } U=[u_1,\dots,u_m]. $$ What I wish is to have a closed-form solution by some form of decompositon of matrix $V=[v_1,\dots,v_m]$. But as I calculated the solution in the case $m=2$, I found that it is very complex and I am now wondering if this question has a closed-form solution.