Suppose $A$ is a commutative algebra over a field $k$.
It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the module of Kahler differentials. By definition, it is a module over $A$ generated by symbols $da,a\in A$ satisfying
$dc=0$ if $c$ is "constant", i.e. $c\in k$ viewed as a subset of $A$.
$d(a+b)=da+db$
$d(ab)=(da)b+a(db)$
$(da)b=b(da)$
There is also a map $d\colon A\to \Omega^1_{k}(A)$, $d(a):=da$ called the de Rham differential.
It is well-known that $\Omega^1_{k}(A)$ and $d$ can be defined by a universal property. Recall first that a map $\phi\colon A\to M$ for some $A$-module $M$ is called a derivation of $A$ with values in $M$ if $\phi(ab)=\phi(a)b+\phi(b)a$. Then $\Omega^1_{k}(A)$ and $d$ are characterized by the property that for any derivation $X\colon A\to M$, there exists unique morphism $\mu_X\colon \Omega^1_{k}(A)\to M$ such that $X=\mu_x\circ d$. You can read all this in details here.
My question is the following. Is there a similar universal description of the de Rham differential $d^1\colon \Omega^1_{k}(A)\to \Omega^2_{k}(A)$? What about $d^n\colon \Omega^n_{k}(A)\to \Omega^{n+1}_{k}(A)$? I would like to see a description like this:
$d^1\colon \Omega^1_{k}(A)\to \Omega^2_{k}(A)$ is a map satisfying some properties, such that any other map $\Omega^1_{k}(A)\to M$ satisfying these properties factors through $d^1$.
Thank you very much for your help!
You can describe the universal property of all of the de Rham differentials at once as follows. The direct sum $\Omega(A) = \bigoplus_i \Omega^i(A)$ together with the differential has the structure of a graded-commutative dg-algebra. There is a forgetful functor from graded-commutative dg-algebras to commutative algebras sending a dg-algebra to its degree-$0$ subalgebra, and the functor $A \mapsto \Omega(A)$ is its left adjoint.