This answer isn't immediately obvious reading the text of Shapechanger, so it's worth going through point by point.
First, the changeling's "true form" is humanoid, has flesh, and is not a skeleton.
Next we look at the relevant abilities that Shapechanger provides (Eberron: Rising from the Last War, p.18):
- Coloration, hair length, and sex
- Height and weight (to a degree)
- Appearance as a member of another race (there's some restrictions, but they aren't relevant here)
The first won't help. You're changing these attributes relative to your current appearance: as noted, changelings start out roughly humanoid, and none of these can remove the things that make a person not a skeleton. The second is a bit more promising, in that you could become emaciated, perhaps even skeletal. Not particularly satisfying, but the option is there.
The last option is a little more ambiguous. You might be thinking that you could take on the appearance of the skeleton monster: after all, it's a creature that is humanoid in shape, statted similarly to other races in the Monster Manual. Unfortunately, "race" is defined in the Player's Handbook to be "one of the many intelligent humanoid species in the D&D world." (PHB p.11) The skeleton monster is neither intelligent, nor humanoid (by the game's standards.)
So it appears that the best a changeling could do would be to appear dangerously thin.
You still have options
To present a different option than the changeling, have you considered the warforged from the same source book? Much of the warforged's description talks about how customized their appearance can be, and as artificial beings, they share more in common with a grim-reaper-style archetype than a typical humanoid.
For example, warforged do not need to eat, sleep, drink, or breath. Their construction prominently features a framework of some kind of hard material, analogous to a skeleton. The illustrations of them carry notably skull-like faces. Many embrace a concrete purpose, which would work well with your grim reaper duties (if you're looking for function in addition to appearance.)
Some modify themselves, or even seek out artificers to do so for them. It's well within the realm of possibility that a warforged could have been constructed as, or choose to appear as, something approaching a skeleton.