Assume that the matrix $A$ is Hermitian/skew-Hermitian. There is a theorem, which says that if this matrix represents some transformation $T$ with respect to an orthonormal basis, then this transformation itself is Hermitian/skew-Hermitian.
What happens if the basis of the representation of $A$ is NOT orthonormal?
The theorem I know does not hold. However, can we still call such a matrix Hermitian/skew-Hermitian since it does not imply that the transformation it represents is the same type (Hermitian/skew-Hermitian)? And if so, does it have any uses?