Not really. Yes, they're quite sensitive, but they throw out a tremendous amount of information (in terms of the neural impulses sent to the brain, it's nothing like the output of a microphone) which means the brain must do a tremendous amount of interpretation/interpolation to try and fill in the missing information.
And how can you actually measure the output from the eardrum, to the anvil and hammer (bones), to the nerve cells in the inner ear, to the brain? Everyone's nerve cells don't pick up all the same frequencies because some nerve cells get damaged over time. When you finally build a microphone with an ear shaped exterior, a diaphragm that is the same shape and made from the same muscle fibers attached to identically shaped bones as those in our own ears, and can accurately replicate an inner ear and translate the nerve signals into electric signals that can be measured, then we will get a full picture of what the ear is capable of relaying. Until that point, I don't see how you can since it's limited by damaged nerve cells and/or varying tone maps from person to person. And what about Fletcher-Munson curves? That has to be taken into account as well.