I am interested in how I could express $\Omega^k( M \times N)$ in terms of $\Omega^i(M)$ and $\Omega^j(N)$ for $i,j = 0,1, \ldots k$. Is there a nice relation?
This question arose in the context of the Freund-Rubin solution to the bosonic equations of motion for 11-dimensional supergravity, where one posits a space-time geometry $AdS_4 \times S^7$ and a 4-form field strength proportional to the volume form on $AdS_4$, considered as an element of $\Omega^4(AdS_4 \times S^7)$. In verifying that such a geometry/field strenth do indeed satisfy the equations of motion, it is necessary to consider its Hodge dual in $\Omega^7(AdS_4 \times S^7)$. I suspect that this is simply the volume form on $S^7$, but can't show this.
I also suspect that the answer to my more general question concerning $\Omega^k( M \times N)$ is not necessary to determine the special case in the second paragraph, but I am quite curious.