I am trying to figure out how the orientation forms on $S^2$ constitute as generators for $H^2_{dR}(S^2 \times S^2)$. I've calculated that $H^2_{dR}(S^2 \times S^2) = \mathbb{R} \oplus \mathbb{R} = H^2_{dR}(S^2) \oplus H^2_{dR}(S^2)$ and now trying to figure out how to show that if $\pi_1 : S^2 \times S^2 \to S^2$ and $\pi_2 : S^2 \times S^2 \to S^2$ are the projections and $\omega_1$ and $\omega_2$ orientation forms on each factor $S^2$ then I guess that $\pi_1^\ast \omega_1$ and $\pi_2^\ast \omega_2$ generate $H^2_{dR}(S^2 \times S^2)$?
I tried to look for some liteature on this from John Lee's book as well as other resources, but I did not find any exercise or example showing how this kind of calculations or how orientation forms can be used as generators. Naturally this question generalizes to the question if we have orientable manifolds $M$ and $N$ such that $$H^k_{dR}(M \times N) \cong H^k_{dR}(M) \oplus H^k_{dR}(N)$$ then will the orientation forms on $M$ and $N$ generate the left-hand side.
The Künneth theorem states there is an isomorphism of graded $\mathbb{R}$-algebras $$H^\bullet(M\times N)\cong H^\bullet(M)\otimes H^\bullet(N)$$ In other words, we have $H^k(M\times N)\cong \oplus_{p+q=k}H^p(M)\otimes H^q(N)$. Now, if you have two vector spaces $V$ and $W$ with bases $\{e_i\}_{i\in I}$ and $\{f_j\}_{j\in J}$, then $\{e_i\otimes f_j\}_{i\in I,j\in J}$ is a basis for $V\otimes W$. So if you know a basis for $H^\bullet(M)$ and $H^\bullet(N)$, then you get a basis for the tensor product, and thus for $H^\bullet(M\times N)$. All you have to do is pull back and wedge the cohomology classes.
For example, if $M$ and $N$ are compact oriented manifolds with volume forms $\omega_M$ and $\omega_N$, then $\pi_M^*\omega_M\wedge \pi^*_N\omega_N$ is a volume form on $M\times N$, and thus generates $H^{m+n}(M\times N)$. Likewise, if $H^k(M\times N)\cong H^k(M)\oplus H^k(N)$, then it is generated by $\{\pi^*_M\omega_i\}\cup\{\pi^*_N\eta_j\}$ where $\{\omega_i\}$ generate $H^k(M)$ and $\{\eta_j\}$ generate $H^k(N)$.