Does this matrix have signature 0?

Notably, your matrix is equal to $$ M = \pmatrix{0&1\\-1&0} \otimes A $$ where $\otimes$ denotes the Kronecker product. If $A$ has real coefficients, then both the $2 \times 2$ matrix and $A$ have imaginary eigenvalues in conjugate pairs, so that $M$ has signature zero as a consequence of the spectral properties of the Kronecker product.