What is "skew-hermitian"

An n by n complex or real matrix A = (a) is said to be anti-Hermitian, skew-Hermitian, or said to represent a skew-adjoint operator, or to be a skew-adjoint matrix, on the complex or real n dimensional space K, if its adjoint is the negative of itself:
A =  − A
.

Note that the adjoint of an operator depends on the scalar product considered on the n dimensional complex or real space K. If ( ⋅ ∣ ⋅ ) denotes the scalar product on K, then saying A is skew-adjoint means that for all u, v ∈ K one has (Au∣v) =  − (u∣Av) .

In the particular case of the canonical scalar products on K, the matrix of a skew-adjoint operator satisfies $a_{ij} = - {\overline a}_{ji}$ for all 1 ≤ i, j ≤ n.

Imaginary numbers can be thought of as skew-adjoint (since they are like 1-by-1 matrices), whereas real numbers correspond to self-adjoint operators.