You should store your original will in a safe place, along with your other important papers. A good quality fireproof safe is an excellent way to store these documents, and they can be purchased relatively inexpensively. You should let your executor and alternate executor know where your original will and important papers are located, so they can access them and secure them quickly upon your death. It is generally not advisable for most people to store their original will in a safe deposit box at the bank. Your safe deposit box will be sealed by the bank upon your death, so your executor may not be able to access the box in a timely manner after your death.
It is never a good idea to write on or mark up your original will document. Crossing things out and re-writing the language on the original copy of your will could possibly invalidate the document. If you would like to make changes to your will, note the changes on a separate sheet of paper prior to having a new one prepared.