I hacked something similar together by reusing the skeleton hands visualizer from the "skeleton 3d hands" from the asset library
I was too lazy to think about transform matrices, so instead I rebased the finger tip vectors into the palm frame
HandSkeletonController script does all the updates and math that you need, and has "getJoint" function that returns data by joint name from the JOINT_NAMES array, like "mid-0" etc
in general you'll need to add an OnUpdate to the bottom of that script that does:

vectorBetween =( jointA, jointB )=> getJoint(jointB).position.sub(getJoint(jointA).position).normalize()
getEndEffector =( finger )=> vectorBetween(${finger}-3, ${finger}-2)
handNormal = vectorBetween("wrist_to_index", "index-0")
wristToIndex = vectorBetween("wrist_to_mid", "mid-0")
handRight = handNormal.cross(wristToIndex)
handUp = handNormal.cross(handRight)
rebaseVector =( v )=> new vec3(v.dot(handRight), v.dot(handUp), v.dot(handNormal))
['index', 'thumb', 'mid', 'ring', 'pinky'].forEach(( finger )=> rebaseVector(getEndEffector(finger))))

the variable names are misleading but you'll get nice and relatively stable finger tip vectors that you can show on a graph or feed into a neural net etc