In my notes it states that for $3$ linear functionals $ T_1,T_2,T_3 \in V^* $ which is the dual space of $ V,$ one can define the triple product as:
$T_1 \wedge T_2 \wedge T_3= T_1\otimes T_2\otimes T_3- T_1\otimes T_3\otimes T_2+ T_2\otimes T_3\otimes T_1- T_2\otimes T_1\otimes T_3+ T_3\otimes T_1\otimes T_2- T_3\otimes T_2\otimes T_1$
My question, is there away to prove this definition? Or do we just take the definition as granted? Because I also want to show $T_1 \wedge T_2 \wedge T_3 \wedge T_4$, however I cant seem to understand why the above definition holds. Can anyone help me see why it is the case?