Let $X$ be a smooth complex projective variety. Let $\omega_1, \ldots, \omega_n$ be elements of $H^0(X, \Omega^1_X)$ and $\gamma_1, \ldots, \gamma_n \in H_1(X(\mathbb{C}), \mathbb{Q})$. Consider the matrix given by the integrals

$M=(\int_{\gamma_i} \omega_j)$ 

Is it true that $\det(M)=\int_{\gamma_1 \cup \cdots \cup \gamma_n} \omega_1 \wedge \cdots \wedge \omega_n$?

If so could anybody provide a proof or a reference?