Some gothic elements in Sir Edmund Orme include:
1) Women in distress, often exhibiting nervous, overwrought emotions.
It is obvious to the narrator that something is affecting Mrs. Marden's peace of mind when she exhibits strange behavior. At the Parade, the narrator observes that Mrs. Marden 'has obviously had some shock' after Charlotte's return from her walk with Captain Bostwick. On another occasion, with seemingly no reason to, Mrs. Marden spills her tea over the narrator after dinner; she mysteriously hints later that she has 'intuitions.'
2) The presence of the supernatural.
It is obvious that Mrs. Marden is terrified of Sir Edmund Orme's appearance; Charlotte does not see him until he appears at her mother's deathbed. Indeed, Mrs. Marden confirms to the narrator that Sir Edmund Orme is dead and has come back to inflict suffering on her for cruelly discarding him to marry Captain Marden years ago. The ghost of Sir Edmund Orme has come back to make sure that Charlotte, Mrs. Marden's daughter, does not exhibit the irresponsibly coquettish ways of her mother in her youth.
The mother was to pay, in suffering, for the suffering she had inflicted, and as the disposition to jilt a lover might have been transmitted to the daughter, the daughter was to be watched, so that she might be made to suffer should she do an equal wrong.
3) An atmosphere of mystery and suspense.
The suspense leading up to the finale is a classic gothic element in fiction. With Charlotte's assent to the narrator, will Sir Edmund Orme's ghost now be pacified that Charlotte will not capriciously throw over a good man (in this case, the narrator) for another (Captain Bostwick)?
Was the sound I heard... the despairing cry of the poor lady's death-shock or the articulate sob (it was like a waft from a great tempest), of the exorcised and pacified spirit? Possibly the latter, for that was, mercifully, the last of Sir Edmund Orme.
4) The presence of rival lovers; an uncertainty of reciprocation.
The narrator has to contend that his beloved, Charlotte, also has another admirer in Captain Bostwick. Typical of gothic stories, he does not know whether he will be successful in his suit until the story culminates with the death of Mrs. Marden. Charlotte is at once coquettish as well as temperamental; it is obvious that she enjoys all the attention her suitors lavish on her. It is intimated that Sir Edmund Orme's ghost is a ghoulishly compelling presence and that he will not rest in spirit until Charlotte does what he deems is the right thing.
There was something different in her, different from all the past. She had recognised something, she felt a coercion. I could see that she was trembling.
Hope this helps!