The skew-adjoint convective derivative operator

Many CFD papers talk about using the skew-symmetric form of the convector operator in the Navier-Stokes equations. The math behind this terminology has confused me for some time (e.g. the operator's nonlinear in the Navier-Stokes equations), so finally I dug through the algebra to show why and when the linear operator is skew-Hermitian and/or skew-adjoint.

The skew-adjoint (skew-symmetric) form of the convective derivative