Let $A=k[x_1,\dots,x_m]$ be the algebra of commutative polynomials in $m$ variables over a field (or commutative ring) $k$, of arbitrary characteristic.  Denote by $B=\bigwedge(x_1^*,\dots,x_m^*)$ the exterior algebra in $m$ variables $x^*_1,\dots,x^*_m$ dual to $x_1,\dots,x_m$.  Let $C=B^*$ be the dual vector space to $B$; it is naturally a coassociative coalgebra over $k$.  In other words, $C$ is the exterior coalgebra cogenerated by the variables/elements $x_1,\dots,x_m$.

The algebra $B$ is naturally augmented with the augmentation morphism $b:B\to k$ taking the generators $x_1^*,\dots,x_m^*$ to zero.  Let $B_+\subset B$ denote the augmentation ideal, i.e., the kernel of $b$.  Passing to the dual coalgebras, we have the coaugmentation morphism $c:k\to C$ with the cokernel $C_+=B_+^*$.  The vector space $B_+$ is naturally an associative algebra without unit, and $C_+$ is a coassociative coalgebra without counit.

Assuming that $A$ is ungraded, or in other words, when viewed as a graded algebra it sits entirely in degree $0$, the algebra $B$ should be graded so that its generators $x_j^*$ sit in the cohomological degree $1$.  Then the cogenerators $x_j$ of the coalgebra $C$ sit in the cohomological degree $-1$.  Now we are doing the cobar construction, so the cohomological grading on $C_+$ will be shifted forward by $1$, making the elements $x_j\in C_+$ sit in the cohomological degree $0$.

The cobar construction $Cob(C)$ is the free associative algebra generated by the cohomologically graded vector space $C_+[-1]$.  The differential $d$ on $Cob(C)$ is induced by the comultiplication in $C$ (with some sign rule).  This should be the smallest possible choice of a cofibrant replacement of the algebra $A$ in the model category of associative DG-algebras over $k$.

In the case of $m=2$, the DG-algebra $Cob(C)$ is the free associative algebra with two generators $x_1$ and $x_2$ in cohomological degree $0$ and one generator $x_{12}$ in cohomological degree $-1$.  The differential is provided by the rule $d(x_{12})=x_1x_2-x_2x_1$. Generally for an integer $m\ge1$, the DG-algebra $Cob(C)$ is a free associative algebra with $2^m-1$ generators.