Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} \xrightarrow{d^1} \Omega^2_{A/R} \to \dotsc$. The construction is given as an exercise in Lang's Algebra (XIX, Theorem 3.2). Somehow this is not completely formal since $\Omega^i_{A/R}$ only has a universal property as an $A$-module, not as an $R$-module, right?
We depand that $d^{p+q}(\omega \wedge \eta) = d^p(\omega) \wedge \eta + (-1)^p \omega \wedge d^q(\eta)$ for $\omega \in \Omega^p, \eta \in \Omega^q$. This implies $d^1(a * d^0(b)) = d^0(a) \wedge d^0(b)$, and more generally $d^1$ maps $\sum_i a_i * d^0(b_i) \mapsto \sum_i d^0(a_i) \wedge d^0(b_i)$. Conversely, one could try to define $d^1$ this way. But then it is not clear to me how to show well-definedness. I will sketch what I have done to remedy this, but probably it's way too complicated, and I wonder if there is a more direct approach.
There is an isomorphism $\Omega^1_{A/R} \to I/I^2$ mapping $a * d(b) \mapsto [ab \otimes 1-a \otimes b]$, where $I$ is the kernel of the multiplication map $A \otimes_R A \to A$. It fits into a long exact sequence $\dotsc A^{\otimes {n+1}} \to A^{\otimes n} \to \dotsc$ where the differentials are alternating sums, for example $A^{\otimes 4} \to A^{\otimes 3}$ maps $a \otimes b \otimes c \otimes d$ to $ab \otimes c \otimes d - a \otimes bc \otimes d + a \otimes b \otimes cd$. Exactness implies that $I$ is isomorphic to the cokernel of this differential. Now define $A^{\otimes 3} \to \Omega^2_{A/R}$ by $a \otimes b \otimes c \mapsto d^0(ac) \wedge d^0(b)$. This is a well-defined $R$-linear map. One checks that it vanishes on the image of $A^{\otimes 4} \to A^{\otimes 3}$, hence extends to $I$. One then checks that it also vanishes on $I^2$, so that it extends to $I/I^2 \cong \Omega^1_{A/R}$. The resulting map $d^1$ has the desired description.
Once one has $d^1$, one might define the other differentials inductively by $d^{p+1}(\omega \wedge \eta) = d^p(\omega) \wedge \eta + (-1)^p \omega \wedge d^1(\eta)$. But then again well-definedness seems to be not so clear.
Of course there are no such problems when $\mathrm{Spec}(A)$ is a vector bundle over $\mathrm{Spec}(R)$, which also explains that they don't occur for the de Rham complex on a smooth manifold.
I have found a more simple construction. In fact, there is a universal property of $\Omega^1_{A/R}$ as an $R$-module: It is the cokernel of the $A$-linear (and therefore $R$-linear) map $$A \otimes_R A \otimes_R A \to A \otimes_R A,\quad a \otimes b \otimes c \mapsto ab \otimes c - b \otimes ac - a \otimes bc.$$ Here, the $A$-module structure comes from the last factor in each case.
This alternative construction seems to be quite unknown(?), see also here. The differential is given by $d(a)=[a \otimes 1]$. We exactly mod out the Leibniz rule $d(a \cdot b)=a \cdot d(b)+b \cdot d(a)$.
Now consider the $R$-linear map $A \otimes_R A \to \Omega^2_{A/R},~ a \otimes b \mapsto d(b) \wedge d(a)$. It extends to $\Omega^1_{A/R}$ since $$\begin{align*} & d(c) \wedge d(ab) - d(ac) \wedge d(b) - d(bc) \wedge d(a) \\ &=a \cdot d(c) \wedge d(b) +b \cdot d(c) \wedge d(a) \\ & -a \cdot d(c) \wedge d(b) - c \cdot d(a) \wedge d(b) \\ & -b \cdot d(c) \wedge d(a)-c \cdot d(b) \wedge d(a)\\ &=0 \end{align*}$$
More generally, $\Omega^p_{A/R}$ is an explicit quotient of $A^{\otimes 2p}$, and $A^{\otimes 2p} \to \Omega^{p+1}_{A/R}$ defined by $a_1 \otimes b_1 \otimes \dotsc \otimes a_p \otimes b_p \mapsto d(b_1 \cdot \dotsc \cdot b_p) \wedge d(a_1) \wedge \dotsc \wedge d(a_p)$ preserves the relations (easy calculation as above), so that it extends to $d^p : \Omega^p_{A/R} \to \Omega^{p+1}_{A/R}$.
This construction even generalizes to arbitrary commutative algebra objects in semi-additive cocomplete symmetric monoidal categories. In the abelian case we can define de Rham cohomology. Details appear in my thesis in the section about derivations.