In my notation, [...] is the same as Alt$\,\equiv\frac{\textstyle 1}{\textstyle k!}\sum\,sgn(\pi)\,.\,.\,.\,$. Now, my answer to the question:

There are various equivalent ways of introducing the wedge product. One possibility is to first define $\,\wedge\,$ for unit forms,
$$
e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_r}\;\equiv\;r!\,\left[\,e^{i_1}\,\otimes\;.\,.\,.\;\otimes\,e^{i_r}\,\right] \quad,\qquad
$$
and then to employ the simple theorem
$$
\left[\,e^{j_1}\otimes\, .\,.\,.\,\otimes\,e^{j_{k+l}}\,\right]\;=\;
\left[\;\;
\left[\,e^{j_1}\otimes\, .\,.\,.\,\otimes\,e^{j_k}\,\right]\;\otimes\;\left[\,e^{j_{k+1}}\otimes\, .\,.\,.\,\otimes\,e^{j_{k+l}}\,\right]
\;\;\right]
$$
as a means to extend the definition of $\,\wedge\,$ to arbitrary skew forms.

To this end, we rewrite the above equality as
$$
\left[\;\;\left(\,\;\frac{\textstyle 1}{\textstyle k!}\;\,e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_k} \,\;\right)\,\otimes\,
\left(\,\;\frac{\textstyle 1}{\textstyle l!}\;\,e^{i_{k+1}}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}} \,\right)\;\;\right]\;=\;
\frac{\textstyle 1}{\textstyle (k+l)!}\;\;e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}}
~\qquad
$$
and then as
$$
\frac{(k+l)!}{k!\;l!}\;\left[\;\;\left(\,\;e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_k} \,\;\right)\,\otimes\,
\left(\,\;e^{i_{k+1}}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}} \,\right)\;\;\right]\;=\;
e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}}
~\;\;.\qquad\qquad
$$
We now see that, if we extend the definition of $\,\wedge\,$ to arbitrary exterior forms as
$$
\omega^k\,\wedge\,\omega^l\;\equiv\;\frac{(k+l)!}{k!\;l!}\;\left[\;\omega^k\,\otimes\,\omega^l\;\right]\;\;
$$
and apply it to $\;\left(\,\;e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_k} \,\;\right)\;$ and $\;\left(\,\;e^{i_{k+1}}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}} \,\right)\;$, we end up with
$$
\left(\,\;e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_k} \,\;\right)\,\wedge\,
\left(\,\;e^{i_{k+1}}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}} \,\right)\;=\;e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_{k+l}}\;\,.
$$
This serves as a motivation to introduce the factor of $\,{(k+l)!}/(k!\,l!)\,$ in the above definition.

To draw to a close, the purpose of the factorials is to make the operation $\,\wedge\,$ associative.

quotientof tensor algebra by the relation $\alpha\wedge \alpha=0$. Forcing it inside the tensor algebra is ugly and unnatural. $\endgroup$2more comments