HKR in characteristic p

Question

Let $A$ be a commutative $k$-algebra where $k$ is a field. The HKR states that $HH_i (A) \simeq \Omega^i_{A/k}$. In characteristic $0$ the isomorphism is given by the formula $a_0 \otimes a_1 \otimes .. \otimes a_n \to \frac{1}{n!} a_0 da_1 .. da_n$ with the inverse given by the antisymmetrization map. My question is: can we write something similar in characteristic p at least for simple examples? For example, for the affine line $\mathbb{F}_p [x]$ we have $HH_1 (\mathbb{F}_p[x]) \simeq \Omega^1_{\mathbb{F}_p[x]}$ but what is the map here? The map from char $k$=$0$ case doesn't work (even for $n=1$) since $a \otimes x^p$ goes to $adx^p = 0$.