The cohomology of $\text{Gr}_2(\mathbb{R}^{n,1})$
A formula for $H^{\ast,\ast}(\text{Gr}_k(\mathbb{R}^{n,1});\underline{\mathbb F}_2)$ appears in my article RO(C2)-graded cohomology of equivariant Grassmannian manifolds, in New York Journal of Mathematics,
preprint available here.
If we express partitions as Young diagrams, it makes sense to define the "trace" of a partition. Now define $\text{part}(p,k,m,t)$ to be the number of partitions of p into at most $k$ natural numbers, none exceeding $m$, which have trace $t$. Then
$\displaystyle H^{\ast,\ast}(\text{Gr}_k(\mathbb{R}^{n,1}))=\bigoplus_{p\le k(n-k)\atop q\le p}\Sigma^{p,q}\mathbb{M}_2^{\oplus \text{part}(p,k,n-k,q)}$
In particular,
$
\begin{align*}
\displaystyle H^{\ast,\ast}(\text{Gr}_2(\mathbb{R}^{n,1}))&=\bigoplus_{p\le 2(n-2)\atop q\le p}\Sigma^{p,q}\mathbb{M}_2^{\oplus \text{part}(p,2,n-2,q)}\\
&=\mathbb{M}_2\oplus
\left(\bigoplus_{p\le 2(n-2)}\Sigma^{p,1}\mathbb{M}_2^{\oplus \text{part}(p,2,n-2,1)}\right)\oplus
\left(\bigoplus_{p\le 2(n-2)}\Sigma^{p,2}\mathbb{M}_2^{\oplus \text{part}(p,2,n-2,2)}\right)\\
\\
&=\mathbb{M}_2\oplus
\left(\Sigma^{1,1}\mathbb{M}_2\oplus\bigoplus_{p=2}^{n-2}\Sigma^{p,1}\mathbb{M}_2\oplus\Sigma^{n-1,1}\mathbb{M}_2\right)\oplus
\bigoplus_{p=4}^{2(n-2)}\mathbb{M}_2^{\oplus \text{part}(p,2,n-2,2)}\\
\end{align*}
$