As with most aspects of music, music theory doesn't offer one single 'correct' way to think of chords. The most commonly-taught way is to start with the triads that are generated from the set of notes in major and minor keys, and then consider other forms as variations on, or substitutions for, those chords. As others have mentioned, there's also an emphasis on looking at how chords function within the context of a progression. These two ideas are deeply ingrained in Western musical tradition, but of course you don't have to think this way - there's no reason you have to think of triads as the basic building block, or think of chords as 'altered' forms of other chords for example. You also don't have to think of chords in terms of their function - many pieces (even by canonical Western composers) arguably use chords primarily for their particular colour and feel ahead of any considerations of function.
You might wish to study some of the theories of consonance, dissonance, and roughness, which might help you if you decide to study harmonies that are generated from the full range of continuous pitches, rather than just the chromatic scale. (I mention this because you said you are interested in 'all permutations' - but you could also 'go bigger' than that idea!). This might also be relevant if you want to study how different chords can sound very different when played with instruments of different timbres.
If restricting yourself to a discrete scale, as Dom says, pitch class set notation is a way of viewing all possible sets of notes in a more egalitarian way. As Sova mentioned, as you stray further from diatonic norms, the 'letters' and 'sharps and flats' way of naming things arguably becomes less useful.