Characters and finite group representations

Question

I just saw the proof of the theorem where 2 representations of a finite group $G$ with the same characters are isomorphic\equivalent representations. The proof uses the orthogonality relations of characters for irreducible characters, and immediately deduce the fact that if we have all the irreducible characters in both reps the exact number of times, then we have equivalence of the representations. I could not see why this proof is sufficient. Could anybody elaborate more about it? I would expect the proof to give explicitly the map which we use to show that the representations are equivalent. But I could not see such proof anywhere in the literature. What am I missing in this proof?