Dual problem of a maximization primal problem $P$?
As you mention, maximization and minimization problems are equivalent. Given a recipe for dualizing a minimization problem, you can convert it into a recipe for dualizing a maximization problem as follows:


Negate the objective function and change the maximum to a minimum.
Compute the dual.
Negate the objective function and change the maximum to a minimum.


The new dual problem will have the same objective value as the primal one (under some mild conditions). This property easily follows from the same fact for duals of minimization problems.

If you're careful enough, you should also get that dualizing twice brings you back to the original program.