The equation you have written is technically correct. It's just that in stoichiometry of ionic compounds, we often use the empirical formula - it is simpler and doesn't really make a difference as what we're really looking for is relative amounts of each chemical.
The overall equation is
Net Ionic Equations:
I'm not sure what you meant by crossing, but both the magnesium ion and barium ion have charges of 2+. That means the anions they exchange are also the same charge.