I personally like the following approach.
Define the tensor algebra $T^{\circ}(V)=\bigoplus V^{\otimes n} = k\oplus V \oplus V\otimes V +\cdots$. Because there natural isomorphisms $V^{\otimes m}\otimes V^{\otimes n} \cong V^{\otimes m+n}$, and tensor products distributes over direct sums, we have a natural multiplication on $T^{\circ}(V)$. Because tensor products are associative, this multiplication is associative. Because we have a natural isomorphism $k\otimes_k V \cong V$, the multiplication is unital. Further more, because tensor products (over $k$) are $k$-linear, all the maps in sight are $k$-linear. Thus, $T^{\circ}(V)$ is a noncommutative, associative, unital $k$-algebra.
There is a grading on $T^{\circ}(V)$ where $V^{\otimes n}$ has degree $n$, and so $T^{\circ}(V)$ is graded. If we quotient out by a homogeneous ideal, the result will still be a noncommutative, graded, associative, unital $k$-algebra. So let's quotient out by the (two sided) ideal generated by all elements of the form $v\otimes v$ where $v\in V$. We call the quotient ring the exterior algebra of $V$ and denote it $\bigwedge^{\circ}(V)$. You can actually define the product to be the wedge product, and you get all the properties (except skew symmetry) from the properties of $T^{\circ}(V)$. Skew symmetry comes from the defining relations in the ideal we quotiented by.
So we can get all the basic properties of the wedge product without passing to combinatorics. Of course, you still have to link this definition of wedge product with whatever other definition you are using, and since I don't know your definition, I can't really say if there will be any combinatorics involved in that.
Unfortunately, we have replaced combinatorics (which, while messy, is elementary) with tensor products and graded rings, which are slightly higher level. In particular, this is probably not an appropriate definition for an undergraduate class in calculus on manifolds, at least one that doesn't presuppose some algebra.