Yes, you have to edit the skin ASCX file. I your scenario above do this:

Find you skin file, say it's called skin1.ascx. copy it and name it skin2.ascx

Assuming you have applied skin1 as you skin for the entire site, On your login page go to page settings and set the page skin to skin2.ascx

Now open skin1.ascx in visual studio, html editor or text editor. Locate the tag that says <dnn:search ........ />. Just remove, save and you're done. That may mess up the look and feel of your page so you may have to play around with the layout to get it right.

It's whatever skin you have assigned to it.  Open the page properties, Appearance and look at the skin.  You will find the skins at /portals/_default/skins/{SkinName}/{Skin}.ascx.  The /_default/ portal might be a portal number if the skin is installed for only a specific portal.

You want to remove the Search object, a line that will look something like:

<dnn:SEARCH runat="server" id="dnnSEARCH" .....