$g(n).h(n)$ will be $\Omega (n)$
Since, we know that $g(n) = \Omega (n)$, so it can be any function which is asymptotically greater than or equal to n,
i.e $g(n) \geq c_{1}n$.
And, $h(n) = O(n)$, i.e. $h(n)$ can be any function which is asymptotically less than or equal to n,
i.e. $h(n) \leq c_{2}n$.
When we multiply both these functions, we can only conclude that it will be $\Omega (n)$.
Because $h(n)$ can possibly be $O(1)$.
And since there is no upper bound on $g(n)$, so we cannot put any upper bound on $g(n)h(n)$.
Hope that helps :)