(For the purposes of this question, please identify each natural number with the set of all smaller numbers. For example, $3 = \{0,1,2\}$.)
I've been thinking about writing up an elementary account of $k$-vectors and wedge products that I can give to students in their last year of high school or their first year of university. For example, I'd like to teach determinants and cross-products in this way. I had in mind something like this: we define $\Lambda^r\mathbb{R}^n$ (we could even denote this $\Lambda^{r,n}$) as the set of functions $$f:(r \rightarrow n) \rightarrow \mathbb{R}$$ satisfying the following condition.
Axiom. For all permutations $\pi : r \rightarrow r$ and all functions $\varphi : r \rightarrow n$, we have $f(\varphi \circ \pi) = f(\varphi)\mathrm{sgn}(\pi).$
For example, if $f$ satisfies the above axiom, then for each $\varphi : r \rightarrow n$ that fails to be injective, we have $f(\varphi)=0$. To see this, let $\pi$ denote a permutation that transposes two elements on which $\varphi$ returns the same value, we have:
$$f(\varphi) = f(\varphi \circ \pi) = f(\varphi)\mathrm{sgn}(\pi) = -f(\varphi).$$
Hence $f(\varphi)=0$.
Anyway, as I was toying with this, I suddenly had a vague recollection of seeing this kind of thing in an algebraic topology subject I took a couple of years ago; it had something to do with abstract simplicial complexes.
Question. Am I recalling correctly that there's a notion/definition/construction similar to this (and probably more general), that's used in algebraic topology in connection with abstract simplicial complexes?