The cohomology of $\text{Gr}_2(\mathbb{R}^{n,2})$
A formula for $H^{\ast,\ast}(\text{Gr}_2(\mathbb{R}^{n,2});\underline{\mathbb F}_2)$ appears in my article RO(C2)-graded cohomology of equivariant Grassmannian manifolds, in New York Journal of Mathematics,
preprint available here.
$\displaystyle H^{\ast,\ast}(\text{Gr}_2(\mathbb{R}^{n,2}))=
\mathbb{M}_2\oplus \Sigma^{1,1}\mathbb{M}_2\oplus \Sigma^{2,1}\mathbb{M}_2$
$\displaystyle \qquad\qquad\qquad\qquad \oplus\Sigma^{2,2}\mathbb{M}_2\oplus (\Sigma^{3,2}\mathbb{M}_2)^{\oplus 2}\oplus (\Sigma^{4,2}\mathbb{M}_2)^{\oplus 3}$
$\displaystyle \qquad\qquad\qquad\qquad \oplus\bigoplus_{i=5}^{n-2}(\Sigma^{i,2}\mathbb{M}_2)^{\oplus 2}\oplus\Sigma^{n-1,2}\mathbb{M}_2$
$\displaystyle \qquad\qquad\qquad\qquad \oplus \Sigma^{5,3}\mathbb{M}_2\oplus\bigoplus_{i=6}^n(\Sigma^{i,3}\mathbb{M}_2)^{\oplus 2} \oplus \Sigma^{n+1,3}\mathbb{M}_2$
$\displaystyle \qquad\qquad\qquad\qquad \oplus \bigoplus_{i=8}^{n+1}(\Sigma^{i,4}\mathbb{M}_2)^{\oplus\lceil\frac{i-7}2\rceil}$
$\displaystyle \qquad\qquad\qquad\qquad \oplus \bigoplus_{i=n+2}^{2n-4}(\Sigma^{i,4}\mathbb{M}_2)^{\oplus (n-1-\lceil\frac{i}2\rceil)}$