Can I send email without an email address using Simple Email Service?

Previously I have used the PHP mail() function to send emails from an app hosted on a Debian Linux / Apache server which was not with AWS. I did not need to have an email address in order to send those emails from the app. The app is only in development stage so I do not have extensive feedback about it. The emails do get sent so it does work at some level.

I am now moving the app to an AWS EC2 Ubuntu Linux server and the mail() function is not working. I realize I have quite a lot of choices about how to get it working. As far as I can see, if I use SES, I will need to have an an actual email address which would be the address that the emails sent from my app would be sent from (previously they were sent from a non existent address). Is that correct? Is it a bad idea to send emails from an app which are not associated with a real email address?