For oriented manifolds $M$ and $N$, I want to give $M\times N$ an orientation. (*) Additionally, I want to show that this orientation makes positively oriented bases $(v_1, \dots , v_m)$ and $(w_1, \dots , w_n)$ for $T_pM$ and $T_qN$ give a positively oriented basis $(v_1 \times 0, \dots , v_m \times 0, 0 \times w_1, \dots , 0 \times w_n)$ for $T_{(p,q)}(M \times N) = T_pM \times T_qN$.
Let $\{(U_i,\varphi_i)\}_{i\in I}$ be an oriented atlas for $M$ and $\{(V_j,\psi_j)\}_{j\in J}$ an orientable atlas for $N$. Then $M\times N$ has product atlas generated by the charts $\{(U_i\times V_j,\varphi_i\times \psi_j)\}_{i\in I,j\in J}$. These charts cover $M\times N$. Since the charts on $M$ and $N$ are orientable, it follows that for each $p\in U_i\cap U_j$ and $q\in V_i\cap V_j$ that the Jacobian matrices $d(\varphi_j\circ\varphi_i^{-1})_p$ and $d(\psi_j\circ \psi_i^{-1})_q$ have positive determinant.
But then $$\det(d((\varphi_j\times\psi_j)\circ (\varphi_i\times \psi_i)^{-1})_{(p,q)})=\det(d(\varphi_j\circ\varphi_i^{-1})_p\times d(\psi_j\circ\psi_i^{-1})_{q})=\det(d(\varphi_j\circ\varphi_i^{-1})_p)\det(d(\psi_j\circ\psi_i^{-1})_q)>0$$
What I don't know is how an oriented manifold (defined in terms of an oriented atlas, that is an atlas generated by a choice of covering charts, where all chart transitions have positive determinant jacobian matrices) induces an orientation on each tangent space.
Do we just fix an oriented basis $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in I}$ for the $n$-dimensional manifold $M$, and for each $p\in M$ where $U_\alpha\ni p$ take $\varphi_\alpha^*\omega$ for $\omega$ the standard orientation form on $\Bbb R^n$, which is well defined since chart transitions about $p$ preserve orientation.