Given the permutation matrix $P\in\mathbb{R}^{n^2}$ such that $P\mathop{\mathrm{vec}}(X)=\mathop{\mathrm{vec}}(X^T)$ for all $X\in \mathbb{R}^{n\times n}$, is there any closed form expression for the eigenvectors of $P$? Clearly, the matrix is symmetric as $P_{m,n}=P_{n,m}^T$ for general vectorized transpose matrices, and as such it must have real eigenvalues $\lambda_i \in\{-1,1\}$, thus if we find a basis for one subspace, the other quickly follows.
EDIT: Towards answering my own question, the -1 eigenspace is equivalent to a basis for $n\times n$ skew-symmetric matrices and the 1 eigenspace is a basis for symmetric matrices. The simplest basis for either is $v=\mathop{\mathrm{vec}}(\pm E_{ij}+E_{ji})$ for $i<j$, plus the diagonal basis $w=E_{ii}$ for the symmetric case, where the former can normalized by multiplication with $\sqrt{2}/2$.