prove or refute (in either case I'll credit you if I use it)
\begin{prop}
Given a $3\times3$ real matrix $\bf A$ with singular values
$\{1,\sigma^2,0\}$, there exist a unique factorization
\[
    {\bf A} = {\bf R}_u \,\, \text{diag}(1, \pm\sigma^2, 0) \,\, {\bf R}_v^t
\]
where 
\[
{\bf R}_u=({\bf I}-{\bf T}_u)({\bf I}+{\bf T}_u)^{-1}
\quad
{\bf R}_v=({\bf I}-{\bf T}_v)({\bf I}+{\bf T}_v)^{-1}
\]
and ${\bf T}_u$ and ${\bf T}_v$ are skew-symmetric matrices. 
\end{prop}

A general $3\times3$ skew-symmetric matrix $\bf T$ is the
cross-product matrix defined as
\[
    {\bf T}= 
    \left(\begin{matrix}
    0 & -t_z & t_y \\
    t_z & 0 & t_x \\
    -t_y & -t_x & 0
    \end{matrix}\right)
\]

% Hint: accordind to SVD a factorization does exit. But SVD is not unique