This is too long for a comment...

Your first statements are not always correct. For three or more particles, one must proceed with extreme care because there are presentations of $S_3$ or $S_n$ that are not 1-dimensional, and neither symmetric nor antisymmetric: a state can be partially symmetric and not return to a multiple of itself after every permutation.

To construct antisymmetric states, one must combine the multidimensional spatial and spin states of representations that transform by conjugate representations of $S_n$.   For $S_2$ is collapses to symmetric and antisymmetric representations, respectively but not for $n>3$ or greater.  See for instance

> Harvey M. [The symmetric group and its relevance to fermion physics][1]. CM-P00067619; 1981.

In particular, it is clearly not possible to create fully antisymmetric states of three spin-1/2 particles, so in such a case multidimensional irreps of $S_3$ (or more generally of $S_n$) will have to be combined.  In particular the spin and space parts of the states are no longer separable.

This is a very inefficient way, and one usually prefers [Slater determinants][2] of space and spin states, which are automatically antisymmetric, as a construction method for fermionic states.


  [1]: https://cds.cern.ch/record/135395/files/CM-P00067619.pdf
  [2]: https://en.wikipedia.org/wiki/Slater_determinant