So I calculate the character map for $ S_5 $, and the only thing I have not figured out yet is how we know the character values of the 6-dimensional irrep line. The irreps I have are: trivial, sgn, standard and the product of standard and sgn. I then calculated the dimensions of the remaining irreps using elementary number theory. If I can calculate the line for 6-dimensional irrep, the rest is not difficult in column orthogonality.
The sources that I have found somehow cite the values or say that it is the "outer square of the standard representation", which we have not discussed in class; he). Is there another way to see this irrep?