Yeah the key step is to find a class on the total space which generates the cohomology of each fiber ---- when you're looking at a vector bundle E-> B and its associated projective bundle PE-> B, there is a tautological line bundle on PE which restricts to the tautological bundle on each fiber which does the trick (since, of course, its first chern class is a generator of the cohomology of P^n)
In this case I think taking the universal bundle on BU(n-1) (or what's it called - the associated bundle?) does the same trick too