Haskell: Can typeclasses define types (ala type traits)

Question

Is it possible to have a type be part of a typeclass? Something like:

class KeyTraits v where
    keyType :: *
    key :: v -> keyType

data TableRow = { date :: Date, metaData :: String, value :: Int }

instance KeyTraits TableRow where
    keyType = Date
    key = date

And can these "type-level" functions be used elsewhere? For example:

-- automatically deduce the type for the key, from the value type, using
-- the typeclass
data MyMap v = { getMap :: (KeyTraits v) => Map (keyType) v }

I may be doing something completely wrong, but I basically want the ability to define type relationships like the one above (e.g. Certain values already may have data that can be used as a Key). If that's not possible, or is difficult, could you suggest a better design that is more idiomatic?

Thank you!