Dwarfs are horrified by skeletons too unfortunately. This is how I deal with goblin sieges: I have 6 total military squads, each eight dwarfs strong. Two swordsdwarf squads, and one squad each of marksdwarfs, hammerdwarfs, axedwarfs, and speardwarfs. I generally have these squads on a roughly two-month rotation, so four of them are always training and two are always resting/doing civilian labors. After the goblin siege arrives, I send 3 of my most freshly trained squads to dispatch the invaders, then after the filth has been taken care of, I set all but one of my squads to inactive and change their hauling labors to exclusively refuse hauling, while simultaneously disabling refuse hauling for my civilians. In this way, only the military dwarfs have to see the corpses and carry them off to my secluded stockpile, and since they are all soulless battle-hardened killing machines, they don't get any negative or "horrified" thoughts from doing this. After the corpses are safely removed from the area, I set my squads back to training as normal and reclaim all of the goblinite for my soft-tempered civilian dwarfs to retrieve.
As for storing of corpses: I have my corpse refuse stockpile in a corner that no civilians really ever have any need to travel to. Once I figure out how to get to magma, I will make a magma chute and dump them away forever. If a civilian dwarf stumbles upon your pile of bodies, he's likely to drop quickly to "stressed" as seeing all of these dead bodies unfortunately provides the same negative thought as if he had watched them die in front of him. This is something I didn't figure out until after a few goblin sieges and I'm struggling to keep everyone happy now. My butcher, who makes regular trips to my mass grave, is on the verge of going insane.