Is Orthogonal Matrix Symmetric?

All the orthogonal matrices are symmetric in nature. (A symmetric matrix is a square matrix whose transpose is the same as that of the matrix).

Are symmetric matrices orthogonal?

Symmetric matrices with n distinct eigenvalues are orthogonally diagonalizable. since a and b are distinct, we can conclude that v and w are orthogonal.

Is every orthogonal matrix skew symmetric?

In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices.