An orthogonal matrix is a matrix $\hat{M}$ that satisfies:
$$ \hat{M}\hat{M}^\top = \hat{M}^\top\hat{M} = \hat{\mathbb{I}} $$
It has the following properties:
$\hat{M}^{-1} = \hat{M}^\top$
$\mathrm{det}\hat{R} = \pm 1$
Treated as a transformation matrix, it preserves volumes (up to a sign) and inner products.
Its columns, treated as vectors, form an orthonormal basis.
In general doesn't have any real eigenvalues.