Members of the orthogonal group.

Complex analogue of orthogonal matrix.

Applications:

A matrix that equals its transpose:

Can represent a symmetric bilinear form as shown at matrix representation of a symmetric bilinear form, or a quadratic form.

The dot product is a positive definite matrix, and so we see that those will have an important link to familiar geometry.

WTF is a skew? "Antisymmetric" is just such a better name! And it also appears in other definitions such as antisymmetric multilinear map.

where:

When we have a symmetric matrix, a change of basis keeps symmetry iff it is done by an orthogonal matrix, in which case:

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