Is your vector space over the real or complex numbers? That's important because for a space over the real numbers, <x, y> = <y x> while for a space over the complex numbers, <x,y>= <y, x>* where * is the complex conjugate.

If over the real numbers, then <x, y><z, w>= <x, y<z, w>> and, since that is to be equal to < x, T*(w)>, T*(w)= <z, w>y.