My lecture notes say that: ''Given any bilinear form $\tau$ on $V$, there is a uniquely determined linear operator $T$ on $V$ such that $$\tau(v,w) = v \, .(Tw)$$ so once we've fixed a 'starting' bilinear form (the inner product), we can get any other bilinear form $\tau$ from this via a linear operator''.
Why is it necessary to fix a 'starting' point and what does this 'starting' point actually mean? Also is there a way of proving the above?