For $ w \in \bigwedge^j(K) $ and $\mu \in \bigwedge^k(K) $ where $K$ is of dimension $n$, I do understand (or have developed an intuition purely relying on permutations) as to why $$w \wedge \mu = (-1)^{jk}\mu \wedge w$$
However, I am not able to reconcile the signifcance of the negative sign and difference in the exponents (between the above and below expression) in the Leibnitz Rule which dictates that $$d(w \wedge \mu) =dw \wedge \mu+ (-1)^{j}w \wedge d\mu$$
Is there an explanation that solely relies on permutation and combination which can explain the exponent and the negative sign?
(I am able to develop an analogy where the negative can be explained via the following argument. Assume that the $d$ operator has to jump to go the next variable and each jump results in a change of sign. Consequently to reach $\mu$ it has to make $j$ jumps making the sign to alternate $j$ times. However, I don't know whether it can be considered a mathematical explanation for its behavior.)