Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space 
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}\_2
$$
where
$$
\Delta=\{(m,m)\mid m\in M \}
$$
and $\mathbb{Z}\_2$ acts by reversing the coordinates order. 
Then $B(M,2)$ is a (real or almost complex) manifold. Suppose the characteristic classes of $TM$ is given. How to obtain the characteristic classed of $TB(M,2)$? For example,


(1). How to compute the Stiefel-Whitney class
$$
w(TB(\mathbb{R}P^n,2))?
$$


(2). How to compute the Stiefel-Whitney class
$$
w(TB(\mathbb{C}P^n,2))?
$$


(2). How to compute the Chern class
$$
c(TB(\mathbb{C}P^n,2))?
$$



characteristic classes of tangent bundle of 2-nd unordered configuration space
An excellent discussion of symmetric powers in general is given by [Pavle Blagojevich](http://www.maths.ed.ac.uk/~aar/papers/blag.pdf) - one of the first things he mentions is that the $k$-th symmetric power of $\mathbb{R}P^n$ is $\mathbb{R}P^{kn},$ a result due to Arnold. This is NOT true of $\mathbb{C}P^n,$ but for $\mathbb{C}P^\infty$ this is [studied by Gusein-Zade et al](http://arxiv.org/pdf/1103.1562v2.pdf). This might be enough for what you want.(that paper seems quite nice also).


