Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of [Localisation of Modules for a semisimple Lie Algebra in prime characteristic][1], we have the following sentence:

The sheaf $D_X$ of crystalline differential operators on a smooth variety $X$ over $k$ has a non-trivial center, canonically identified with the sheaf of functions on the Frobenius twist $T^∗X^{(1)}$ of the cotangent bundle. Moreover $D_X$ is an Azumaya algebra over $T^∗X^{(1)}$.

Instead of going through the general proof given, I only want to understand, in as simple a manner as possible, the situation when $X$ is the affine $n$-space over $k$. In this case, $D_X$ is simply the Weyl algebra and the Azumaya property, if I understand correctly, means that the quotient of the Weyl algebra by its centre is isomorphic to some matrix algebra over $k$. Is there a way to construct such a matrix algebra and a corresponding isomorphism to the quotient explicitly? Any help, even in the case of the affine line, would be highly appreciated.

**P.S.** If my understanding is incorrect, could you please point out the flaw(s) and how the question could be turned into something reasonable?

  [1]: https://arxiv.org/pdf/math/0205144.pdf