Best way to store tomato paste/puree?
We buy large cans at home just like you do. We prevent it from spoiling by freezing it. Stored in the freezer the paste stays good for months, it's just a matter of correct storage.

What we do is the following: spoon a portion into a plastic bag -> twist the bag around several times -> tie off with a tierib (we use the thingies you get in the package with the bags themselves) and then repeat until done.

Every time you want to use the paste, just cut off a portion from the sausage-like string you will have.

Other options include using icecube trays, but we don't prefer that due to the small size of the cubes and the fact that the trays don't have a lid.