BEFORE you pay the principal down, check two things. Call your lender/servicer/bank (the number on your statement), and ask these questions:

Is there a pre-payment penalty on my loan that will be acitvated by a partial bulk payment of principal?

Do you offer a "recast" option for borrowers making a bulk principal reduction?

You certainly do not want to trigger a pre-payment penalty on your loan by paying down early. Some lenders will consider a 20% or more reduction of principal within the pre-payment period as the same as "pre-payment" of your loan. The details of your pre-payment penalty will be in your note (which you should have received at closing), but sometimes the details of partial pre-payment are not spelled out. Check this before you do ANYTHING ELSE.

Second, ask the lender if they will "recast" your new, lower balance. Many lenders will take the remaining principal after a reduction of 20% or more, and re-amortize or re-calculate your payments based on the new, lower amount. Generally they want you to be current with a good history of payment, and pay a processing fee of a couple hundred dollars to accomplish this. Occasionally, they will want to run an automatic valuation of your property. Each lender has their own process or procedure for this, but with most it is fairly straight forward. Best if they can provide the new terms and a summary of the process in writing (many do).

A minority of lenders, particularly portfolio or niche lenders, will NOT offer this option. They maintain that if you pre-pay principal, it will simply reduce the time that it takes to pay off your loan, but will not alter your payment. If your lender is one of these, you will be unable to reduce your payments without obtaining a NEW loan and paying off the old one - that is, refinancing.

If you are forced to refinance in order to lower the payment, carefully evaluate the cost for feasibility - and definitely check at least one independent source along with your current lender. If your current lender knows your intentions and is offering no solution but a refinance, they may also suspect that you are a captive audience and not offer the most competitive rate. I would love to say that is rare, but I suspect it is not as rare as one might imagine - so do your due diligence for rates and terms up front.

If your loan is a standard amortized loan, the pay down of the balance does not automatically lower the house payment requirement, it simply reduces the balance and then of course the interest cost and the term of the loan, but not the monthly payment. 

You certainly can contact the lender that is holding the loan, (Servicer) and ask for a modification of the payment (re-amortize).  Typically if the reduction is somewhat substantial in comparison to the overall balance, they will approve this modification. Sometimes a fee will be charged of possibly $250.  It would be better to ask for the modification prior to the principal payment.

Understand that the servicer of the loan is not typically the owner of the loan, so they may not have the ability to make that decision on their own, but would need to ask for permission from the loan holder.

This is called "re-cast" or "re-amortize".  This is a relatively common practice and pre-payment penalties do not typically apply, although I would ask to make sure.  Call the servicer of your mortgage loan and explain what you want to do.  At that point they will direct you on the procedure.  They typically charge a  small fee to do this.  Usually under $500.  They should be able to accept the payment and give you your new payment information.  This is not a "regular" practice, but it is fairly common, but whoever you are speaking to, make sure they understand exactly what you are trying to accomplish.  Once you are satisfied, go ahead and move forward.