No, you can only deduct cash expenses that you pay as an employee. Your expenses have to be pretty high in order to claim any of them, because first you'd have to be able to itemize deductions (have expenses greater than $6300 for a person filing single), and your employee expenses would have to exceed 2% of your Adjusted Gross Income.

The mileage expense that you may have heard about refers to automobile expense, which is a simpler way to deduct the costs (gas, insurance, etc) of using a car for business.