Excellent observation, as it identifies how the settings of Poe's stories reflect the characters of their protagonists. Whether it is the catacombs in "The Cask of Amontillado" that reflect Montresor's disturbed and twisted character of the House of Usher, whose dilapidation reflects the own mental instability of its owner, Roderick Usher, Poe always uses setting to great effect in his gothic, spine-chilling tales of horror. Consider what we are told about the House of Usher:
Its principal feature seemed to be that of an excessive antiquity. the discoloration of ages had been great. Minute fungi overspread the whole exterior, hanging in a fine tangled web-work from the eaves. Yet all this was apart from any extraordinary dilapidation. No portion of the masonry had fallen; and there appeared to be a wild inconsistency between its still perfect adaptation of parts, and the crumbling condition of the individual stones. In this there was much that reminded me of the specious totality of old woodwork which has rotted for long years in some neglected vault, with no disturbance from the breath of the external air.
Note the overall emphasis on rottenness and decay. The setting of course is a symbol that could be said to represent the madness and mental disturbance at the heart of the owner of the house. At first glance it appears to be of sound quality, but closer examination reveals issues that could indicate serious structural problems. Remember, the narrator has journeyed here, not out of choice, but because Roderick Usher, his childhood friend, has written to him to come and be with him as he is suffering from a "nervous agitation" and a "mental disorder". How mentally disturbed he is will only be revealed at the end of the story... Your question correctly identifies that in a sense the house is a symbol of the diseased and rotting character who owns it.
The setting of the Fall of the House of Usher serves as a character because the house itself is described thoroughly from top to bottom in a way that clearly shows the mental, psychological, and emotional state of the house's owner, Roderick Usher. Using this description, the reader can understand that there had been a moment in the life of the character where things were fine, grandiose and even wonderful. Yet, somewhere along the line, something supernatural (and we know genetic and inevitable) has brought down the line of Usher with disease, despair, and destruction. This is why, at the first description that we find, we see how the narrator is shocked to see the detrimental state of the house. Shortly after in the story, he will use similar descriptors to refer to Usher himself.
I looked upon the scene before me—upon the mere house, and the simple landscape features of the domain—upon the bleak walls—upon the vacant eye-like windows—upon a few rank sedges—and upon a few white trunks of decayed trees—with an utter depression of soul which I can compare to no earthly sensation more properly than to the after-dream of the reveller upon opium—the bitter lapse into everyday life—the hideous dropping off of the veil. There was an iciness, a sinking, a sickening of the heart—an unredeemed dreariness of thought which no goading of the imagination could torture into aught of the sublime What was it—I paused to think—what was it that so unnerved me in the contemplation of the House of Usher? It was a mystery all insoluble; nor could I grapple with the shadowy fancies that crowded upon me as I pondered. I was forced to fall back upon the unsatisfactory conclusion, that while, beyond doubt, there are combinations of very simple natural objects which have the power of thus affecting us, still the analysis of this power lies among considerations beyond our depth.