Signature form of $S^2 \times S^2$

Question

Let $M=S^2 \times S^2$ be the product of two copies of the $2$-sphere. We have that $dim(M)=4$. So we can define the intersection form $$ I_{S^2 \times S^2} := H^2(M, \mathbb{Z}) \times H^2(M, \mathbb{Z}) \to \mathbb{Z} $$ that makes the assignament $(c_1, c_2) \mapsto \langle c_1 \cup c_2, [M] \rangle$, where with $ \cup$ I denote the cup product in cohomology. How can I conclude that the form $I_{S^2 \times S^2}$ can be represented with the matrix $H= \begin{pmatrix} 0 &1 \\1 & 0 \end{pmatrix}$ ? I know the geometric motiv: I can take a generator of $H^2(M)$ that is $S^2 \times pt$ and it has not self-intersections and it has an intersection with the other generator. But what is the algebric way in order to solve the problem?