General linear group on manifold

Question

Knowing that $\mathrm{O}(n,\mathbb{R})$ is a closed submanifold (of the general linear group) and that $\mathrm{SO}(n,\mathbb{R})$ is one of its subgroups with the same dimension, is there a quick way (possibly using only basic arguments, i.e. without any reference to Lie theory) to see that $\mathrm{SO}(n,\mathbb{R})$ is a maximal connected subset of $\mathrm{O}(n,\mathbb{R})$ ?