How to construct a WebSocket URI relative to the page URI?

If your Web server has support for WebSockets (or a WebSocket handler module) then you can use the same host and port and just change the scheme like you are showing. There are many options for running a Web server and Websocket server/module together.

I would suggest that you look at the individual pieces of the window.location global and join them back together instead of doing blind string substitution.

var loc = window.location, new_uri;
if (loc.protocol === "https:") {
    new_uri = "wss:";
} else {
    new_uri = "ws:";
}
new_uri += "//" + loc.host;
new_uri += loc.pathname + "/to/ws";

Note that some web servers (i.e. Jetty based ones) currently use the path (rather than the upgrade header) to determine whether a specific request should be passed on to the WebSocket handler. So you may be limited in whether you can transform the path in the way you want.