As I noted in my preceding question https://math.stackexchange.com/questions/3510189/give-a-general-class-to-which-a-specific-4-times-4-special-orthogonal-matrix in equation (62) of their recent publication https://arxiv.org/abs/1708.05336, "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}

For $n=3, 5, 6$, I have tried (via direct enumeration) unsuccessfully to construct analogous $n \times n$ special orthogonal matrices, in which all the (equal) entries of the last column and row are (for probabilistic reasons) positive, and the remaining $n^2- 2n +1$ entries are all equal in absolute value. Such matrices might be helpful in extending the Li-Qiao framework to the construction of separable decompositions of length $n \neq 4$. (It is not clear, however, that matrices must be of the specific requested form to so extend their framework. Perhaps, other than having the last row and columns positive, all remaining entries could be unrestricted, other than for the orthogonality requirement.)

It was observed by Robert Israel in the noted preceding question that Q is proportional to a Hadamard matrix. However, the next larger-sized ($8 \times 8$) Hadamard matrices are not orthogonal in character, so this does not seem to be a productive direction to take. (But as the comments below of others and mine indicate I was in error in making this claim.)