Go to Beyond-a-century.com and buy their BHT powder, it is a fat-soluble antioxidant. Mix into a new bottle of flax oil, about 1/4 teaspoon and it will protect the oil from oxidation. BHT is pretty cheap, something like 8 bucks for 300 grams.
Shaking up the bottle of flax oil speeds up the rate of oxidation, due to the air you are trapping in the oil. This process is not immediate, but happens slowly over time. Likewise, mixing up a shake with flax oil will not destroy the oil, this is a gradual process.