Ser Lipman's Asterisque book on dualizing sheaves and differentials who works over a perfect field and provides a canonical isomorphism.

Ser Lipman's Asterisque 117 entitled "Dualizing sheaves, differentials and residues on algebraic varieties" who works over a perfect field and provides a canonical isomorphism.

Addendum: I wrote the answer in a hurry, I apologize. Let me be more explicit. In the book, working with a variety $X$ over a perfect field $k$, Lipman constructs a certain sheaf $\omega_X$ (actually a sheaf on the big Zariski site over $Spec(k)$) called the canonical sheaf, by using rational differentials and traces, together with a canonical map

$$c_X \colon \Omega^n_X \to \omega_X$$

Then he proves two things

1. The sheaf $\omega_X$ is dualizing, i.e. it represents the functor $H^d(X,-)^\vee$, where $(-)^\vee$ denotes $k$-dual and $d$ the dimension of $X$.
2. If $X$ is smooth over $k$ the map $c_X$ is an isomorphism.

The map $c_X$ is called the fundamental class and admits a big generalization using sheafified Hochschild Homology, but this is another story.