I'm not too comfortable with "why" questions, but one
One good reason for the ubiquity of the exterior algebra construction is that it has nice universality basic properties . As Ben mentioned, the exterior algebra of a (super)vector space which if made precise will uniquely define it):
I can give a very incomplete description of the distinguished nature of fermions
Bonus properties in the resulting state vector should differ only by multiplication by finite rank case include:
In this determinant (and pretty much any) setting, symmetric and antisymmetric behavior is distinguished as pointed out by the fact that any nontrivial symmetric group has only two possible one-dimensional complex representations. Bosons correspond to the trivial representation - switching does nothing to the stateMarc Nieper-Wißkirchen).Fermions correspond to the sign representation - switching multiplies by -1. The space of states for all finite collections of identical bosonic particles (called the bosonic Fock space) is therefore the symmetric algebra on the single-particle space, and the fermionic Fock space is
Regarding the exterior algebra.
When it comes to pointing out ways the exterior algebra is useful when the symmetric algebra is not, I think most all of the applications you listed revolve around the finite rank properties - in particular, the distinguished nature of the determinant as a canonical one-dimensional tensor. The only one-dimensional symmetric tensor is the trivial one, which carries no information. Any attempt to make things like volumesboth multilinear and nontrivial , cup products, or Hodge stars requires an orientation, which can be viewed as a determinant (see e.g., my answer here).
As Wikipedia mentions, exterior algebras satisfy a universal property: for any linear map from a vector space $V$ to an associative algebra $A$ landing in the square zero subspace, there is a unique algebra homomorphism from $\bigwedge V$ to $A$ making a certain triangular diagram commute. This yields a description of the exterior algebra functor as a left adjoint to a forgetful functor from strictly supercommutative algebras. If 2 is invertible, then it is equivalent to the parity-shifted symmetric algebra functor, but in general, it represents a genuinely different functor. In particular, the fact that determinants exist in characteristic 2 is an indication that the exterior algebra is more important than the shifted symmetric algebra.
I should emphasize that supersymmetry is different from the existence of fermions. In short, fermions are just odd fields that transform a certain way under ordinary spacetime symmetries, but supersymmetry is the specification of additional odd symmetries of spacetime. This is substantially more exotic: theories containing fermionic particles (like electrons) can exist without supersymmetry, and sign considerationsin fact existed happily for about 50 years before supersymmetry was hatched.