TITLE: Examples of matrices that are both skew-symmetric and orthogonal
QUESTION [7 upvotes]: Are there matrices that satisfy these two conditions? That is, a matrix $A$ such that
$$A^T=A^{-1}=-A$$
What I know is that a skew-symmetric matrix with $n$ dimensions is singular when $n$ is odd.

REPLY [5 votes]: A general classification: Note that any such matrix is normal, hence unitarily diagonalizable. Any orthogonal matrix is unitary, so all of it's eigenvalues are contained in the unit circle $C=\{z:|z|=1\}$ of the complex plane. Similarly any skew-symmetric has eigenvalues in $i \Bbb R$, i.e., purely imaginary. Thus the orthogonal, skew-symmetric matrices are precisely those matrices whose eigenvalues lie in $C \cap i \Bbb R = \{-i,i\}$, and which are unitarily diagonalizable.
In odd dimensions, the characteristic polynomial has at least one real root, hence no such matrices exist. In even dimensions, such matrices are precisely those whose eigenbasis is orthonormal, and whose characteristic polynomial is $(z-i)^{d_1}(z+i)^{d_2}$ where $d_1+d_2=d$. If we want the matrix to have real coefficients, then necessarily $d_1=d_2=d/2$, and thus the characteristic polynomial is $(z^2+1)^{d/2}$. This implies (by orthonormal diagonalizability), that the matrix mentioned by @Alex is the only real skew symmetric and orthogonal matrix, up to an orthogonal change of basis (i.e. conjugation in $O(d, \Bbb R)$).