Orthogonal maps of $\mathbb R^{2n+1}$ have an eigenvalue $\pm1$

A linear transformation T is called orthogonal if it is nonsingular and \({ } ^ { t } T = T ^ { - 1 }\) . Prove that if \(T : { \mathbf { R } } ^ { 2 n + 1 } \to { \mathbf { R } } ^ { 2 n + 1 }\) is orthogonal, then there exists a vector \(\boldsymbol { v } \in \mathbf { R } ^ { 2 n + 1 }\) such that \(T v = \pm v\)