You may be interested in Hal Clement's novel The Nitrogen Fix as a reference.
In that setting, organisms do not use gaseous N2O as a metabloic oxidizer exclusively, but generally make use of a range of nitrogenous compounds in solid, liquid, and gaseous form. This is based on extrapolation from the existing biological nitrogen cycle on Earth--lots of organisms produce nitrogen oxides because nitrogen is important biological material for building amino acids, etc., and nitrites and nitrates are considerably more useful as raw materials for other anabolic processes than diatomic nitrogen is... and conversely, plenty of organisms break down nitrogen oxides, using them as electron receptors (i.e., oxidizers) in their metabolism, because they can. Nitrous oxide in particular is the least thermodynamically favorable electron acceptor out of the variety of nitrogen oxides, but it is nevertheless produced as an intermediary in both the creation and destruction of nitrates and nitrites, and organisms do then use it as an oxidizer in the final stage of reducing nitrogen oxides back to water and diatomic nitrogen.
Natural processes that generate nitrous oxide are part of nitrification and denitrification cycles. Specifically, they include:
- Stepwise autotrophic oxidation of ammonia to nitrite (NO−2 + water) and nitrite to nitrate (NO−3) (nitrification).
- Stepwise reduction by facultative anearobes of nitrate to nitrite, nitrite to nitric oxide (NO), and nitric oxide to N2O and ultimately N2.
In autotrophic NH3-oxidising bacteria, both of these processes can occur, resulting in the pathway NH3 + 2 O2-> NO−2 + 2 H+ + 2 H2O (oxidation), followed NO−2 + 2 e + 2 H+ -> NO + H2O (reduction), and 4 NO + 4 e + 4 H+ -> 2 N2O + N2 + H2O.
So, purely as a matter of biochemistry, yes, dinitrogen monoxide could in fact be used as an oxidizer for life. What reactions would be needed to produce it? Probably similar reactions as are used on Earth for nitrogen fixation: initial reduction of diatomic nitrogen to ammonia, followed by a cycle of re-oxidation and further reduction. It is, however, conceivable that some organism could develop an enzyme for directly attaching a free oxygen radical to diatomic nitrogen as a way of cleaning up after itself during oxygen photosynthesis or something, to avoid polluting its environment with toxic reactive oxygen.
What reactions would be needed to utilize it? Exactly the ones used on Earth: let it accept two electrons, and the oxygen will detach and hook up with solvated protons to from water and leave behind diatomic nitrogen.
The real question is, why is life on this world producing so much nitrous oxide, without producing any nitric oxide or nitrogen dioxide, both of which are easier to produce from ammonia and both of which are much more broadly useful? Perhaps autotrophic nitrogen-fixing bacteria-analogues use of all of their own nitrate and nitrite to power their own metabolism, but don't bother with using N2O as an oxidizer because it's less efficient, so it builds up for other organisms to figure out how to use?