Exterior-power Lie algebra representation ($x\cdot (v_1\wedge \cdots \wedge v_r)={\sum }_i{v}_1\wedge \cdots \wedge x{v}_i\wedge \cdots \wedge v_r$)

A Lie algebra representation on $V$ induces one on each exterior power by acting on every factor and summing. The tensor product of Lie algebra representations preserves the defining alternating relations, so this descends to the quotient in every characteristic of a field. In characteristic two define the exterior algebra by the relations $v\wedge v=0$, rather than by dividing a tensor antisymmetrizer by $2$.