  We prove that the Grassmannian of totally isotropic $k$-spaces of the polar
space associated to the unitary group $\mathsf{SU}_{2n}(\mathbb{F})$ ($n\in
\mathbb{N}$) has generating rank ${2n\choose k}$ when $\mathbb{F}\ne
\mathbb{F}_4$. We also reprove the main result of Blok [Blok2007], namely that
the Grassmannian of totally isotropic $k$-spaces associated to the symplectic
group $\mathsf{Sp}_{2n}(\mathbb{F})$ has generating rank ${2n\choose
k}-{2n\choose k-2}$, when $\rm{Char}(\mathbb{F})\ne 2$.
