The torsion $\Theta:=\mathcal{D}\theta$ of a connection one-form $\omega$ for a principal $G$-bundle with respect to a solder(ing) form $\theta$ is given by $$\Theta=d\theta+\omega\wedge\theta$$ where $d$ is the exterior derivative and $\mathcal{D}$ is the exterior convariant derivative.
The Bianchi identity for torsion is given by $$\mathcal{D}\Theta=\Omega \wedge\theta$$ where $\Omega$ is the curvature of the connection one-form.
My question is: in both of these statements, how is $\wedge$ defined? $\omega$ is a Lie-algebra-valued one-form and $\theta$ is a $V$-valued two-form, where $V$ is the representation space of a linear representation of the Lie group $G$, so how can we wedge these things together?
To make this more specific still (and to make clear my motivation for asking), I'd like to understand how the "wedge" product explained at 1:07:25 here https://www.youtube.com/watch?v=j36o4DLLK2k is defined; I'm struggling to understand Dr Schuller's explanation.