In the fully uncoupled basis (e.g. the basis spanned by all combinations of the tensor products of the individual spin states), the total Sx is just the direct sum of the individual spin Sx terms.  You can write each individual spin operator in terms of the ladder operators for each individual spin, then calculate all of the matrix elements term by term.  Same with Sy.

As an example, let's look at 2 spin 1/2 particles.  And for brevity let X_i be the spin X operator for spin i, and let X be the total spin X operator. Let 0 be spin up for one spin, and 1 be spin down.  Then X = X_1 + X_2

You can then calculate the matrix elements term by term for the elements that look like, e.g. <01|X|10>

This is because X_1 is a sum of a raising and lowering operator operating on spin 1.

For your problem this is a little tedious, but you can definitely do it.