How would you go about balancing this chemical equation?

You have a typo, there is no CO3.

When a metal carbonate decomposes, you get the metal oxide + carbon dioxide, so I really think the equation you're looking for is CaCO3 ----------> CaO  +  CO2
But that's already balanced.

It's the CO3 on the product side I was talking about.  What I should have said is that there is no neutral CO3 molecule.

There is no reaction between CaCO3 and O2. There is no compound with the formula CO3.
If you heat CaCO3 - in the air or anywhere- it decomposes. It does not react.
The products of the decomposition are CaO and CO2. The balanced equation is:
 CaCO3(s) ? heat ? CaO(s) + CO2(g)

You ask about CO3 = carbonate: Yes: CO3 2- is a carbonate ion with a 2- charge. It cannot exist as a compound.

Tips:whatever eqn in your left hand side should always be the same at your right hand side.For example decomposition of calcium trioxocarbonate (iv) CaCO3 will give CO2 +CaO you see here when balanced we still have the same number of oxygen,2 moles of oxygen in the CO2 plus one mole of oxygen in the CaO equals the number of oxygen in the original equation which was 3.The eqn is balanced bcos 1 mole of calcium  in the LHS is equal to 1 mole in the RHS,1 mole of carbon in the LHS is still equal to 1 mole in the RHS,and 3 moles of oxygen in the LHS is s till the same in the RHS(O2+O=3).N:B LHS means left hand side while RHS means right hand side respectively.Tips:whatever eqn in your left hand side should always be the same at your right hand side.For example decomposition of calcium trioxocarbonate (iv) CaCO3 will give CO2 +CaO you see here when balanced we still have the same number of oxygen,2 moles of oxygen in the CO2 plus one mole of oxygen in the CaO equals the number of oxygen in the original equation which was 3.The eqn is balanced bcos 1 mole of calcium  in the LHS is equal to 1 mole in the RHS,1 mole of carbon in the LHS is still equal to 1 mole in the RHS,and 3 moles of oxygen in the LHS is s till the same in the RHS(O2+O=3).N:B LHS means left hand side while RHS means right hand side respectively.