Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring 
$$
H^*(F(\mathbb{R}P^n,k);R)$$
is obtained for any commutative ring $R$ with unit and $2$ invertible. 
I want to find $$
H^*(F(\mathbb{R}P^n,2);\mathbb{Z}).$$

When I use the Serre spectral sequence for the fibration $$
\mathbb{R}P^n\setminus *\simeq \mathbb{R}P^{n-1}\to F(\mathbb{R}P^n,2)\to \mathbb{R}P^n,
$$
I do not know how to determine the differentials. How can I get $$
H^*(F(\mathbb{R}P^n,2);\mathbb{Z})?$$

Are there more general results for $$
H^*(F(\mathbb{R}P^n,k);\mathbb{Z})?$$