In this document by Keller, proposition 2.1, it is stated that for every $A_\infty$-algebra $A$ there is a universal dg-algebra $U(A)$ w.r.t. the existence of an $A_\infty$-morphism $A\to U(A)$, and this universal morphism is a quasi-isomorphism.
Now, what is $U(A)$? My first thought (I haven't understood rectification properly yet) would have been to quotient out the images of all higher structure maps $m_n$, $n> 2$ of $A$, but I am not sure if that gives any meaningful object.
Is there a hands-on description of the dg-algebra Keller mentions?
As Keller says, "cf. Section 4.8". In Section 4.8, he says "Let us put $U(A) = \Omega B_\infty A$." Here $\Omega$ is the cobar construction (bottom of page 12), and $B_\infty$ is an $A_\infty$-version of the bar construction (defined about line 10 of page 14).