  If $K$ is a field of characteristic $p$ then the $p$-torsion of the Brauer
group, ${}_p{\rm Br\,}(K)$, is represented by a quotient of the group of
$1$-forms, $\Omega^1(K)$. Namely, we have a group isomorphism
$$\alpha_p:\Omega^1(K)/\langle{\rm d}a,\, (a^p-a){\rm dlog}b\, :\, a,b\in K,\,
b\neq 0\rangle\to{}_p{\rm Br\,}(K),$$ given by $a{\rm d}b\mapsto [ab,b)_p$
$\forall a,b\in K$, $b\neq 0$. Here $[\cdot,\cdot )_p:K/\wp (K)\times
K^\times/K^{\times p}\to{}_p{\rm Br\,}(K)$ denotes the Artin-Schreier symbol.
  In this paper we generalize this result. Namely, we prove that for every
$n\geq 1$ we have a representation of ${}_{p^n}{\rm Br\,}(K)$ by a quotient of
$\Omega^1(W_n(K))$, where $W_n(K)$ is the truncation of length $n$ of the ring
of $p$-typical Witt vectors, i.e. $W_{\{1,p,\ldots,p^{n-1}\}}(K)$.
  Explicitly, we have a group isomorphism
$$\alpha_{p^n}:\Omega^1(W_{p^n}(K))/\langle Fa{\rm d}b-a{\rm d}Vb\, :\, a,b\in
W_n(K),\, ([a^p]-[a]){\rm dlog}[b]\, :\, a,b\in K,\, b\neq
0\rangle\to{}_{p^n}{\rm Br\,}(K).$$ Here $F$ is the Frobenius isomorphism, $V$
is the Verschiebung map and $[a]$ is the Teichm\"uller representative of $a\in
K$, $[a]=(a,0,0,\ldots )$.
