When you watch Alex Proyas' original movie of The Crow nowadays, it's an amazingly tight movie. The supernatural revenge plot is pretty straightforward, and the film skips most of the angst, to go straight to the campy fun. But originally, there was a forboding presence in the film: the Skull Cowboy. See for yourself.
Over at Comic Book Movie, the reliably great Nailbiter111 has compiled a superb collection of prop photos, concept art, animated storyboards and deleted scenes about the Skull Cowboy. See some of this stuff below, but a ton more over at Comic Book Movie.
In a nutshell, the Skull Cowboy is a character who originates from the original Crow graphic novel, who provides guidance to Eric, in much the same way as his corvid companion does. Here's one deleted scene from the climax of the film, where the Skull Cowboy warns Eric that he's on the wrong path:
And here's an extended version of Eric's resurrection scene, where the Skull Cowboy is at Eric's graveside:
You can read a draft script for The Crow here, in which the Skull Cowboy pops up in a number of scenes, mostly telling Eric off about his lousy attitude. Saying things like, "You work for the dead. Forget that, and you can forget it all." And, "You need to learn to mind your own business or you'll never get where you think you're going." In a nutshell, the Skull Cowboy lays down rules, mostly having to do with not interfering with the living, and Eric keeps breaking them.
This is a very different sort of story than the atmospheric, weird-as-hell movie we actually got, where Eric seems invincible and fixated on revenge until the bad guys find his Achilles heel towards the end of the film.
Here's a collection of animated storyboards, showing a bunch of Skull Cowboy scenes that were never filmed, including the Skull Cowboy standing at Eric's grave, perching at Eric's loft window, and coming to Eric as he plays guitar to warn him that he can't assist the living, or he'll bleed:
And here's a couple of the great concept art/storyboard pieces by Peter Pound: