You haven't really explained what you're trying to accomplish by putting "events into functions" so it's hard to know precisely what you want it to look like.

At it's heart, nodejs is an event driven environment. As such, you register event listeners and they call you when there's an event. That's the structure of the socket.io connection event.

If you don't want everything inline such as this:

io.on('connection', (socket) =>{
            //code
});

You can put the event listener callback into a function itself such as:

io.on('connection', onConnection);

Then, somewhere else (either in the same module or imported into this module), you would define the onConnection function.

function onConnection(socket) {
    // put your code here to handle incoming socket.io connections
    socket.on('someMsg', someData => {
        // handle incoming message here
    });
}

Note: inline callback functions (of the type you seem to be asking to avoid) are very common in nodejs programming. One of the reasons for using them is that when they are inline, they have available to them all the variables from the parent scope (without having to pass them all as arguments) and there are times this is very useful. So, it might be a style that you will find useful rather than avoiding.

I don't think this is going to structure your code in a way that is more readable and useful to you. But if you did want this, you could use async/await.

async function getNewConnection() {
  return await new Promise((resolve, reject) => {
    io.on('connection', resolve);
  });
}

Then in your other code:

const socket = await getNewConnection();

Really though, for this connection handler you should stick with the callback pattern. It keeps things simple for you. Run your updates when the connection comes in, and not the other way around.

Update: I have kinda found the OOP solution for this (it is not the best):

class ConnectionHandler {

init() {
  this.listenForConnections();
}

listenForConnections() {
 io.on('connection', (socket) =>{
        //do stuff
        this.newConnection = true;
        this.update();
 });
}

// the update method contains all the logical decisions of a class
update() {
   if(this.isThereANewConnection()) {
    //add connection to list.. or handle the rest of the logic
  }
  //render screen
}

isThereANewConnection() {
   if(this.newConnection) {
    this.newConnection = false;
    return true;
  }
 return false;
}

}