Here's what we do....keep in mind I'm on a ICU stepdown/telemetry floor, not a psych unit, and just because someone's depressed doesn't mean they won't also need their appendix out, or have a flare up of their gallbladder...
Put the patient as close as possible to the nurse's station, and keep the door to their room open
Round frequently, but not on a set schedule...if the person knows, "okay, they won't be back for 1 hour" then they have a window of opportunity.
Ask the patient straight up if they want to hurt themselves. Don't sugar coat it. I had a patient who was depressed, and the wife told me he was, I thought he was not only depressed but suicidal, and voiced my concerns to the doc. Doc comes in, "have you ever given some thought to possibly doing yourself some injury..." Guy says "no." I sit down in front of the patient, take his hands, say, "Mr. Smith. Look at me. Do you want to kill yourself?" Guy doesn't answer. "Mr. Smith, have you ever felt that everyone would be better off if you were dead?" He said yes. I then said, "How do you plan on doing it?" He said, "save up my pills." Clear, direct, cut and dried speech. Guy got the help he needed.
Make super sure the pt doesn't have any meds with them, or anything they could use as a weapon. We're not a psych hospital, so there's mirrored glass over the sink, strings on the blinds. When you go in the room, say, "if I wanted to hurt myself in here, how would I do it?" And I've actually had facilities management come in and take the handle off a window, too.
I can't suggest strongly enough that you discuss the 800 pound gorilla in the room. Use the words "kill" and "die," not "hurt" or "injure." I've had suicidal patients before, and I've told them, "killing yourself will not make things better for you. You will spend all eternity feeling like you do right now, this very minute. It won't be better, it won't be a peaceful sleep, you'll feel like this for all time...." The allure of suicide is that the pt thinks it will make the pain stop. Convince them that it won't. I've also told patients, "Look at me. If you kill yourself, I will spend the rest of my life wondering what I could have done to stop you. I don't deserve that. If you feel like you're going to hurt yourself, call me, and I'll be right here. But don't make me feel like I failed you for the rest of my life because you killed yourself...." If I had my way, I'd make them go watch an autopsy. Break the death fantasy of "drifting off to sleep" and replace it with "if you overdose, you will die from drowning in your own vomit."
Finally, get help. Make sure every nurse, every aid knows what's going on, so if he goes into the bathroom and the CNA notices the sheet's missing, she doesn't think it just got dirty and thrown into the laundry. That happened to a friend of mine, and she said the worst thing she ever did was have to leave the noose on the pt's neck for it to be taken off by the coroner...no bath, no clean up, into the bag the way he was, dripping pee and poop. To this day, she'll swap out with someone if we've got a suicidal pt...she never got over it.