Computing the core in a transferrable utility game
It depends on the set of feasible allocations for the coalitions $S$. Suppose for all $S$ a best allocation exists (the sum of the individual utilities of the members of $S$ is maximal). Then as in usual cooperative games each coalition can be assigned this best utility sum as its value $v(S)$. Let us denote the utility vector each player gets from an allocation by $x$, and the sum of utilites players get from this by $x(S)$. An allocation is in the core iff
$$
\forall S\subseteq N: x(S) \geq v(S).
$$ 
This is a slight simplification of what you wrote, because instead of comparing all allocations to $x$ we first find the 'best' allocations for each coalition and only compare these to $x$.

If you make further assumptions about the coalition values $v$, e.g. if you assume the game is convex (this is true for exchange economies) then the calculation may be simplified a bit, as the core of a convex game is the Weber set.