Should I ask him out?

I met a guy at an internship I did over the summer, and he seemed smitten by me from day one itself. His face lights up everytime he sees me or when I say hi to him, and he looks straight down my eyes when talking to me. He also tries to close the distance between us a lot of times by walking up and standing very close to me when we're talking. This one time we were out with our team and during group photographs he always came and stood next to me and our arms brushed but he never stepped back and let the contact remain. Also, he keeps staring at me from everywhere he is at and smiling, and jokes with me every now and then and keeps trying to talk to me. What's more, even after my internship ended he keeps trying to invite me into assignments and day projects. He never leaves without properly wishing me goodbye, and has more than once insisted on dropping me home in his car. I think he's into me, am I mistaken? Should I ask him out because he is kinda shy? Help!