According to Cultureofchemistry.fieldofscience.com, an inch of snow contains about 7 milligrams of nitrogen gas per square foot, or about 1/3 of a kilogram in an acre of snow.
The atmosphere is roughly 80% nitrogen, in the form of N2. The form matters. Nitrogen gas is very unreactive, so much so that it many "air sensitive" materials are packed under pure nitrogen (The part of the air that is reactive is molecular oxygen, O2).
Snow certainly contains dissolved nitrogen gas. Henry's law predicts the solubility of a gas in a solvent, water in this case, as a function of temperature. It might seem at first glance counter intuitive, but gases are more soluble in cold solvents than in warm (the opposite is true of most solids, as anyone who has tried to dissolve sugar in cold ice tea knows).
The trouble is actually that this nitrogen isn't in a form that easily accessible to plants. Nitrogen in the atmosphere must first be "fixed" or changed into a more reactive form, typically tetravalent nitrogen (ammonium) which is then converted to the nitrate ions that plants can use.
You breathe oxygen, yet air is mostly nitrogen. You need nitrogen to live and encounter it in the foods you eat and in many common chemicals. Here are some quick facts about this element shared by Chemistry.about.com.