We usually use constraints/capsules for a skeletal mesh's hit box, but how can we use our own animated collision mesh?
The Physics Asset is exactly what you need. Ignore the capsule component and use the physics asset for your hit box checks.
right click on your physics body, and clock on : "copy collision from static mesh"
Note however that it's better to keep the capsule for movement checks, physics assets for this kind of computation are much heavier. Just use the physics asset with your bullets for example.
Normal collision creation and detection does not work with Skeletal Mesh Actors. In order for your Skeletal Meshes to collide with objects in the level, your Skeletal Mesh Actors need to have a Physics Asset that is specially created for your object.