A social Security number is necessary for tax purposes in the U.S.; without one, I don't think there is an easy way to set up an account.
(There may be a way to do it if you have an Individual Tax Identification Number, or ITIN, but I don't know too much about that.)
If there is a reputable broker in your home country, you may consider investing through them; however, be especially careful who you are dealing with.
Also, since you are just beginning, I would start paper trading and learning more about investing before actually putting real money at risk. Look at websites like wallstreetsurvivor.com or other educational sites (investopedia.com).