Based on my tests so far,  is not just an approximation.  It's right on the money, provided you exclude w(N) from the window function.  I'm talking about its magnitude, not the unimportant linear phase factor.  The symmetry issue for N-even is also unimportant.  It only matters for the academic exercise of trying to create a function centered at n=0 whose DTFT is real-valued.  But most actual windowing is done like Window_function#Window_examples,with an even number of points and a window centered at (N-1)/2; i.e. the visible peak is flat-topped at both w(N/2-1) and w(N/2+1).  So the DTFT generally has linear phase and therefore is not real-valued.