Monogamy to open relationship

do

I’ve been with this guy for about 7-8 months now but apparently we never defined our relationship but I’m very close with his family and is a part of his daily life. He’s been sleeping with other people (which he lied to me about but I let it go) and I haven’t been by my choice. So do you think it will be possible for us to have an open relationship? If we agreed to do so, I could see how it would work but I’m worried I’ll get jealous.