Computing wedge products > Compute $\omega = (e_1^* + e_2^* + \cdots+ e_n^*) \wedge (e_1^* + e_2^*) \wedge (e_1^* + e_3^*) \wedge \cdots \wedge(e_1^* + e_n^*)$ in the standard form. I first thuoght I'd pick a value from the first bracket and wedge it with a value in the next bracket (one I don't already have) and keep doing this and when I do this, I end up getting: $$(e_1^* \wedge e_2^* \wedge \cdots \wedge e_n^*) + (e_2^* \wedge e_1^* \wedge e_3^* \wedge \cdots \wedge e_n^*) + (e_3^* \wedge e_2^* \wedge e_1^* \wedge e_4^* \wedge \cdots \wedge e_n^*) + \cdots + (e_n^* \wedge e_{n-1}^* \wedge e_{n-2}^* \wedge \cdots \wedge e_1^*),$$ but I don't know how to simplify this. But then I realised, I've been wedging the same value more than once. So for example, if I pick $e_1^*$ in the first bracket, the I haved wedged it in the second bracket with $e_2^*$, but then when I pick $e_3^*$, I have wedged that with $e_2^*$ again. So how would I compute this wedge product?

I agree with your calculation except for the last term, which I would have said is $$ e_n^*\wedge e^*_2\wedge\cdots\wedge e^*_{n-1}\wedge e_1^*. $$ Note that each term except the first one in your calculation differs from $e_1^*\wedge\cdots\wedge e_n^*$ by a single transposition, so you get $$e_1^*\wedge\cdots\wedge e_n^*-e_1^*\wedge\cdots\wedge e_n^*-\cdots-e_1^*\wedge\cdots\wedge e_n^*=??$$