void Update()
{
if (Input.GetKey(KeyCode.Escape))
    Screen.lockCursor = false;
else
    Screen.lockCursor = true;
}

Press and hold escape to unlock the mouse.

I've been looking at threads about this problem and haven't found an easy enough answer, so I thought I'd add my two cents:

It's not a perfect solution bot for prototyping purposes I use this:

function Update () {
    Screen.lockCursor = true;
    Screen.lockCursor = false;
}

Screen.lockCursor moves the pointer to the center of the screen, so it does almost exactly what the OP suggested - resets the cursor every frame.

You could also use a MouseMove event for example :)

Not very clean but it does the job, you retain OnMouseDown functionality, the cursor remains visible.

Hello.

I needed what you want during a testing proyect. I used "Screen.lockCursor" to block the cursor in the center of screen to simulate the pointer of a gun.

Here you have the API example: ScreenLookCursor

I hope its what you want. See you.