  We first give a short intrinsic, diagrammatic proof of the First Fundamental
Theorem of invariant theory (FFT) for the special orthogonal group
$\text{SO}_m(\mathbb{C})$, given the FFT for $\text{O}_m(\mathbb{C})$. We then
define, by means of a presentation with generators and relations, an enhanced
Brauer category $\widetilde{\mathcal{B}}(m)$ by adding a single generator to
the usual Brauer category $\mathcal{B}(m)$, together with four relations. We
prove that our category $\widetilde{\mathcal{B}}(m)$ is actually (and
remarkably) {\em equivalent} to the category of representations of
$\text{SO}_m$ generated by the natural representation. The FFT for
$\text{SO}_m$ amounts to the surjectivity of a certain functor $\mathcal{F}$ on
$\text{Hom}$ spaces, while the Second Fundamental Theorem for $\text{SO}_m$
says simply that $\mathcal{F}$ is injective on $\text{Hom}$ spaces. This
theorem provides a diagrammatic means of computing the dimensions of spaces of
homomorphisms between tensor modules for $\text{SO}_m$ (for any $m$). These
methods will be applied to the case of the orthosymplectic Lie algebras
$\text{osp}(m|2n)$, where the super-Pfaffian enters, in a future work.
