In equation (62) of their recent publication, "Separable Decompositions of Bipartite Mixed States", Li and Qiao present the matrix $Q \in \mbox{SO}(4)$,
\begin{equation} Q=\frac{1}{2}\left( \begin{array}{cccc} 1 & -1 & -1 & 1 \\ -1 & -1 & 1 & 1 \\ -1 & 1 & -1 & 1 \\ 1 & 1 & 1 & 1 \\ \end{array} \right). \end{equation}
We see that all the entries of the matrix have the same absolute value ($\frac{1}{2}$), and in the last row and column, all the entries (for probabilistic reasons) are positive. I would be interested in other $4 \times 4$ special orthogonal matrices with these properties, and more broadly still in $n \times n$ special orthogonal counterparts. (I have an application in mind with $n=11$.)