This is a decomposition, no analysis involving coefficients is necessary at all. Convert 0.366 dm3 O2 to a mass of O2 (e.g. using the ideal gas equation) and add this to the mass of your product oxide, and you get the mass of your reactant oxide. If you really want, you can work out how many moles of BaO there were, then take this as the same number of moles as in the reactant to calculate the molar mass of the reactant oxide, after which it will be easy to find the stoichiometric coefficients.