jQuery selector to get form by name

$('form[name="frmSave"]') is correct. You mentioned you thought this would get all children with the name frmsave inside the form; this would only happen if there was a space or other combinator between the form and the selector, eg: $('form [name="frmSave"]');

$('form[name="frmSave"]') literally means find all forms with the name frmSave, because there is no combinator involved.

For detecting if the form is present, I'm using

if($('form[name="frmSave"]').length > 0) {
    //do something
}

You have no combinator (space, >, +...) so no children will get involved, ever.

However, you could avoid the need for jQuery by using an ID and getElementById, or you could use the old getElementsByName("frmSave")[0] or the even older document.forms['frmSave']. jQuery is unnecessary here.

    // this will give all the forms on the page.

    $('form')

   // If you know the name of form then.

    $('form[name="myFormName"]')

  //  If you don't know know the name but the position (starts with 0)

    $('form:eq(1)')  // 2nd form will be fetched.