Below are the types of business entities available. You would not form a partnership because a partnership requires at least two individuals.
The link below provided more info on each entity.
https://ttlc.intuit.com/turbotax-support/en-us/help-article/sole-proprietorship/type-entity-business...
https://www.irs.gov/businesses/small-businesses-self-employed/business-structures
The business entity and the identifying number are two separate things to consider.
Yes, you can use your social security number to file your self-employed income as long as your are not a Partnership, S-corp, C-corp. If you would like to have a identifying number for your business other than your social security number, you would apply for an EIN with the IRS. Although you have an EIN, you would still your report your business income and expenses one Schedule C with your Individual Tax Return.
You can apply for an EIN for free on IRS website.
https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n...
