Explicit examples of alternating multilinear forms I think explicit examples of multilinear forms would help me understand them. (I'm studying determinants.) I know they are helpful (only?) for calculating volumes, which is why any alternating multilinear form $w(x_1,\ldots,x_n)$ vanishes if the arguments are linearly dependent. Also, because they are associated with volumes, it makes sense that switching the arguments will switch the sign. 
What would an alternating multilinear form $w(x_1,\ldots,x_n)$ look like in terms of the components of each vector $x_j = (\xi_{j1},\ldots,\xi_{jn})$ ? 
 A: Let $\omega$ be an alternating two form. We can define $\omega$ by specifying its action on a standard basis. 

$$ \omega(e_1,e_1) = \omega_{11} = 0 $$
$$ \omega(e_2,e_2) = \omega_{22} = 0 $$
$$ \omega(e_1,e_2) = \omega_{12} $$
$$ \omega(e_2,e_1) = \omega_{21}  = - \omega_{12}$$

Now let omega act on  vectors $v = v^1 e_1 + v^2 e_2$ and $w = w^1 e_1 + w^2 e_2$ (note the superscripts here are not powers just labels).
$$ \omega(v,w ) = \omega( v^1 e_1 + v^2 e_2, w^1 e_1 + w^2 e_2 ) $$
$$= v^1 w^1 \omega(e_1,e_1) + v^1 w^2 \omega(e_1,e_2) + v^2 w^1 \omega(e_2,e_1) + v^2 w^2 \omega(e_2,e_2)$$
$$= 0 +  v^1 w^2 \omega(e_1,e_2) + v^2 w^1 \omega(e_2,e_1) + 0$$
$$=v^1 w^2 \omega_{12} + v^2 w^1 \omega_{21}$$
$$=(v^1 w^2  - v^2 w^1) \omega_{12}$$

As you can see the action of the bilinear form can be completely described in terms of its components and the components of the vectors it is acting on. In more classical treatments in tensors you will often encounter expressions like $``$ The tensor $\omega_{ij}$ ". In physics this is still the notation we use today. 

We can see that in general that the components of an alternating form  $\omega_{i_1 i_2 i_3\dots i_n}$ will be zero if any pair of subscripts are the same and antisymmetric under the exchange of any pair of indices. 
Now consider what will happen if we feed this form two vectors which are the same.
$$ \omega( \xi, \xi, \eta, \dots, \zeta ) = \sum_{i_1,i_2,\dots i_n} \omega_{i_1 i_2 i_3 \dots i_n } \xi^{i_1} \xi^{i_2} \eta^{i_3} \cdots \zeta^{i_n} $$
Now we exchange $i_1$ and $i_2$ in $\omega$, picking up a negative sign, and commute the numbers $\xi^1$ and $\xi^2$. 
$$= -\sum_{i_1,i_2,\dots i_n} \omega_{i_2 i_1 i_3 \dots i_n } \xi^{i_2} \xi^{i_1} \eta^{i_3} \cdots \zeta^{i_n} $$
$$ = - \omega(\xi, \xi, \eta, \dots , \zeta)$$ 
So we have that the number $\omega(\xi,\xi,\eta,\dots,\zeta)$ is its own negative. The only way this is possible is if $\omega(\xi,\xi,\eta,\dots,\zeta)=0$.

As an example of a "useful" alternating bilinear form, in Electrodynamics the Electromagnetic Field Tensor $F_{\mu \nu}$ is alternating. 
