  We prove Schur--Weyl duality between the Brauer algebra $\mathfrak{B}_n(m)$
and the orthogonal group $O_{m}(K)$ over an arbitrary infinite field $K$ of odd
characteristic. If $m$ is even, we show that each connected component of the
orthogonal monoid is a normal variety; this implies that the orthogonal Schur
algebra associated to the identity component is a generalized Schur algebra. As
an application of the main result, an explicit and characteristic-free
description of the annihilator of $n$-tensor space $V^{\otimes n}$ in the
Brauer algebra $mathfrak{B}_n(m)$ is also given.
