السَّلاَمُ عَلَيْكُمْ وَرَحْمَةُ اللهِ وَبَرَكَاتُهُ My mother and brothers sleep in one big bed, while I sleep in a separate bed in the same room. If their sleeping arrangement is haram, am I sinning just by sleeping in the same room, as if I am participating in a forbidden gathering?"

No, you are not sinning by being in the room, because sharing one room with each person in his own bed is allowed. As for one bed, once a child reaches ten he is separated in sleeping from his mother, father, brother and sister, so if they have reached that age it should be corrected, and if they are younger than that there is nothing wrong with it.