When we speak of differential forms on a manifold $M$, we have two different visions. One is the point of view of sections of $\Lambda^rT^*M$, the other is of maps from $\mathfrak{X}(M)\times\cdots\times\mathfrak{X}(M)$ to $C^\infty(M)$.
In the first case, the wedge product between two differential forms is clear to me. ($p\in M$ and $X_1,\cdots,X_{r+s}\in T_pM$
$$(\alpha\wedge\beta)(p)(X_1,\cdots,X_r,X_{r+1},\cdots,X_{r+s}):=(\alpha(p)\wedge\beta(p))(X_1,\cdots,X_r,X_{r+1},\cdots,X_{r+s}) \\ = cst\sum_{\sigma\in S_{r+s}}\text{sgn}(\sigma)\alpha(p)(X_1,\cdots,X_r)\beta(p)(X_{r+1},\cdots,X_{r+s})$$
But in the second case, is there a way of defining this wedge product directly? I would like to have a immediate definition for
$$(\alpha\wedge\beta)(X_1,\cdots,X_r,X_{r+1},\cdots,X_{r+s})$$
without having to pass through the other vision. Indeed, one could define it as follows:
$$(\alpha\wedge\beta)(X_1,\cdots,X_r,X_{r+1},\cdots,X_{r+s})(p) = (\alpha\wedge\beta)(p)(X_1(p),\cdots,X_r(p),X_{r+1}(p),\cdots,X_{r+s}(p))$$
But there is maybe a more direct way of defining this.
For example one could imagine
$$(\alpha\wedge\beta)(X_1,\cdots,X_r,X_{r+1},\cdots,X_{r+s}) := \alpha(X_1,\cdots,X_r)\wedge\beta(X_{r+1},\cdots,X_{r+s})$$
since the wedge of $C^\infty(M) = \Lambda^0T^*M$ is well defined.
The answer to your question should be easy. You understand the first definition, don't you? Then copy it to the second case: If $\alpha$ is a totally skew-symmetric map from $\mathfrak X(M) \times \cdots \times \mathfrak X(M) \rightarrow C^\infty(M)$ ($p$ times) and $\beta$ a totally skew-symmetric map from $\mathfrak X(M) \times \cdots \times \mathfrak X(M) \rightarrow C^\infty(M)$ ($q$ times), then you can define
$$ \alpha\wedge\beta(X_1,\dots,X_p,X_{p+1},\dots,X_{p+q}) = \sum_{\sigma\in\mathfrak{S}_{p+q}}\mbox{sgn }(\sigma) \alpha(X_{\sigma(1},\dots,X_{\sigma(p)}) \beta(X_{\sigma(p+1)},\dots,X_{\sigma(p+q)}) $$
In the above definition, you have constructed a totally skew map defined over $\mathfrak X(M) \times \cdots \times \mathfrak X(M) \rightarrow C^\infty(M)$ $p+q$ times and, as you wanted, you haven't use your first definition. I think that is your meaning of ''direct way''.
However, note that the final result of $$ \alpha\wedge\beta(X_1,\dots,X_p,X_{p+1},\dots,X_{p+q}) $$ is a function, and that a way to define a function is saying the value in each point of its domain, i.e.: instead of defining the function in terms of other functions (the right-hand term of my definition), you can define the product point-wise. Therefore you have been recovering your second definition (which you didn't like it a the begining by maybe now you do it).
I hope this answer helps you.