Let $A$ be a complex abelian variety of dimension $d$. Let $\omega_1, \ldots, \omega_j \in H^0(A, \Omega^1_A)$ be linearly independent (so $j \leq d$) and consider $\gamma_1, \ldots, \gamma_j \in H_1(A(\mathbb{C}), \mathbb{Q})$. Look at the matrix

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

Is it possible to express the determinant of $M$ as the integral of $\omega_1 \wedge \ldots \wedge \omega_j \in H^0(A, \Omega^j_A)$ against some element of $H_j(A(\mathbb{C}), \mathbb{Q})$ constructed out of $\gamma_1, \ldots, \gamma_j$?