Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes \begin{equation} \|R-M\|_F \end{equation}

then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are the orthogonal matrices from the singular value decompositon of $M$. i.e. $M = U\Sigma V^{T}$ as explained in this answer. When $UV$ is not a "special" orthogonal matrix i.e. $det(UV^{T}) = -1$ we replace the singular vector $u_3$ by $-u_3$.

My question is why only $u_3$ and why not any other column of $U$ or row of $V^{T}$?

(Posting this as a question instead of a comment since I don't have enough reputation yet, sorry!)