$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$ How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
 A: It is
$${\rm Alt}(\phi_1\otimes\phi_2\otimes\phi_3)=
\phi_1\otimes\phi_2\otimes\phi_3-\phi_1\otimes\phi_3\otimes\phi_2+\phi_2\otimes\phi_3\otimes\phi_1-\phi_2\otimes\phi_1\otimes\phi_3+\phi_3\otimes\phi_1\otimes\phi_2-\phi_3\otimes\phi_2\otimes\phi_1$$
but some other 
will define
$${\rm Alt}(\phi_1\otimes\phi_2\otimes\phi_3)=\frac{1}{6}(
\phi_1\otimes\phi_2\otimes\phi_3-\phi_1\otimes\phi_3\otimes\phi_2+\phi_2\otimes\phi_3\otimes\phi_1-\phi_2\otimes\phi_1\otimes\phi_3+\phi_3\otimes\phi_1\otimes\phi_2-\phi_3\otimes\phi_2\otimes\phi_1).$$
A: You'd write it as
$$\begin{align}
\operatorname{Alt}(\phi_1\otimes\phi_2\otimes\phi_3)
= \frac{1}{6}(&\phi_1\otimes\phi_2\otimes\phi_3
+ \phi_2\otimes\phi_3\otimes\phi_1 + \phi_3\otimes\phi_1\otimes\phi_2\\
&- \phi_1\otimes\phi_3\otimes\phi_2 - \phi_2\otimes\phi_1\otimes\phi_3
- \phi_3\otimes\phi_2\otimes\phi_1)
\end{align}$$
More generally, for $\operatorname{Alt}(\phi_1\otimes\dots\otimes\phi_n)$ you write down the tensor products of all permutations of the $\phi_k$, multiply each of them with the sign of the corresponding permutation, add them up and divide the sum by the number of permutations.
