I used to buy the the tube type tomato paste which is so much easier than the tinned variety. I use tomato paste occasionally and about a teaspoon or so since I cook for just three of us.
What happens when I store it inside the refrigerator is that it starts to mould. I dont like freezing it, Mainly because i have to defrost my freezer once a week or it ends up looking like its snowed in or something.
I have heard of storing pesto with oil layer on top so I tried it on my tomato paste and it worked!!!. So to store , spoon any left over tomato paste into a small bottle that is completely dry, Not even a drop of water should be there even on the lid!!. Use a small teaspoon the flatten the top and then add a thin later of olive oil on top. Put the lid on tight and store in refrigerator. Make sure the tomato paste stays under the oil.
I use this method for both tomato pastes and purees and home made pestos.