With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.

Is there an explicit description of a cobordism between them?

I know the answer for $n = 1$: it reduces to finding a cobordism between $S^2$ and $S^1\times S^1$ (take a solid ball and remove a solid torus from the interior, or take a solid torus and remove a solid ball from the interior).