Reading the definition of the geometric product we get that it satisfies:
$$ a_1 \wedge \cdots \wedge a_r = \frac{1}{r!} \sum_{\sigma \in G_r} \mathrm{sgn}(\sigma)a_{\sigma(1)}\cdots a_{\sigma(r)} $$
Which if I understand properly is $1/r!$ multiplied by the sum of all permutations of the vectors (adjusted for the sign of the perumation).
I am trying to better understand this. My knowledge of the wedge product is basically as a mechanism to construct $k$-vectors, like $e_1 \wedge e_2 = e_{12}$. But this seems to be outputting a scalar?
This confuses me.