Actually the way you describe them in your question is one of the ways, called pitch class sets (http://composertools.com/Theory/PCSets/).
EDIT: Except you use C major as a starting point, which I didn't notice at first. With pitch class sets you use the chromatic scale, so your chords would be (putting C at zero) [0,2,4], [0,5,10], [0,9,17]=[0,9,5].
EDIT2: To classify pitch class sets one can use prime forms. With traditonal third-based chords one can usually reorder the notes in a chord so it forms a tower of thirds, and then use that to decide the quality of the chord. For example, given the notes G,Bb,E,C, one reorders them to C,E,G,Bb, and decides that it's a dominant 7th chord. Obviously one cannot build a tower of thirds (or any other fixed kind of intervals) from an arbitrary pitch class set, so something else must be done.
The idea is to rotate the pitch class set until it is as small as possible. We also consider the inversion (the "write it upside down" kind of inversion) of a set as the "same", since it has the same interval structure and therefore sounds very similar. In particular, every major and minor triad has the same prime form. The basic algorithm to calculate the prime form is (from the link earlier):
- Keep rotating your chord until it is as small as possible.
- If there are ties, then use the rotation that has the notes most packed towards the bottom.
- Check to see if the inversion is better packed.
Let's do this for [0,1,7,9]. The rotations are: [0,1,7,9], [1,7,9,0], [7,9,0,1], and [9,0,1,7]. Of these, [7,9,0,1] is the smallest; no ties. The inversion is [7,8,11,1], which is better packed (more tight on the left). Transposing this to zero, we get the prime form [0,1,4,6].
The prime forms can be ordered and then given a so called Forte number. This answer explains the numbering and Wikipedia has a list of them all. The Forte number of [0,1,4,6] is 4-Z15. Now, just like the quality of the chord G,Bb,E,C is dominant 7th, the quality of C,C#,G,A ([0,1,7,9]) is 4-Z15. Of course, some special chords have more descriptive names. For example, 4-Z15 is one of the two all-interval chords. The dominant 7th chord is 4-27; prime form [0,2,5,8] (which, as you can see, is actually the inversion of a dominant 7th chord).