Prove a transformation is skew-symmetric

Prove a transformation is skew-symmetric

pLet $C(0,1)$ be the real linear space of all real functions defined on
[0,1] with inner product $(f,g) = \int_0^1 f(t)g(t)dt$. Let $V$ be the
subspace of functions such that $\int_0^1 f(t)dt = 0$, and $T: V
\rightarrow C$ be $\int_0^x f(t)dt$. Prove that $T$ is skew-symmetric./p
pI think that I'm supposed to show $\int_0^1 f(t)Tg(t) + Tf(t)g(t)dt = 0$
but I'm not sure how to do it./p