You could if you wanted to, but what if later you wanted to change the connection string for some reason? Youd then have to change all the connection strings in all your forms, then retest them all to make sure they work. Youd also have to do tests to insure that all connections were closed when your app closes.
Referencing an object is a bit faster then creating a whole new one, not to mention it saves a little bit of memory. But who really cares about that in this day and age? You can really just do what you want. Its your program and your design, Im just tossing out possible problems that it may cause later.
Using a single module (as your first post mentions) to provide a public property for your connection is also a valid choice, as is Robbys suggestion of putting it inside a class too (which makes more sense for .NET).
Bottom line is, do what you want. Just try and make a good decision and think ahead so your programs are easy to update at a later date. All I was doing was offering an alternative, and to be honest Robbys suggestion is probably better then mine. I originally misread the initial post which is why I offered an addition suggestion.