A standard small bale of hay weighs between 40 and 75 lbs. It is intended to be handled without machinery. You can find larger bales that are 100 lbs or more, but those aren't for one person to huck around by plain muscle power. These days, they make huge round bales, that are more like 1200 lbs.
I suspect you mean the first of these. If it isn't falling from a significant height, it may not do any damage at all, since it isn't terribly hard.
Short answer: I'd go with 1d6 per 10' fallen general rule (assuming a modern 40 lbs bale)
Long answer: It also depends whether we're talking about an actual hay bale or a straw bale, which people tend to confuse. Then it depends how "historically accurate" you want to be. Baling requires some kind of press and mechanical equipment, which may or may not be present in your (presumably) fantasy world. The level of technology of this machinery would make the stack tighter or looser, bigger o smaller, in turn affecting the expectation of damage in D&D. Early balage produced enormous but not particularly tight bales of 500-800 lbs if memory serves.
Even large amounts of loose hay can hurt mind you. I would keep the 1d6 per 10' fallen (or 10' worth of) rule for an avalanche of loose hay.
More bonus situations:
Light it on fire first before dropping it? +1d6 fire damage = Hay Fever
Deliver your attack line broodingly blandly? +1d6 psychic damage = Hay-den Christensen
Shake it like a Polaroid picture? DC 13 Wis save or suffer effect of Otto’s Irresistible Dance = Hay Ya