Here's the definition.
Let T be a linear operator on a finite-dimensional inner product space V. If [itex] \|\vec{T(x)}\| = \|\vec{x}\| \\[/itex] for all x in V, we call T a unitary operator.

The question is asking about for all x in some orthornormal basis for V. Isn't that the same as for all x in V?

Pick ANY orthonormal basis. Call it {e_1,e_2,...,e_n}. You want to define U somehow. The only condition that U has to satisfy is that ||U(e_i)||=1 for 1<=i<=n.