Source: http://www.mathwords.com/a/area_convex_polygon.htm
What kind of determinant is that? Is there a standard convention to interpret determinants in that form?
Of course, looking at the expansion, one can guess that the same determinant could be written as a sum of $2\times2$ determinants but the I am concerned about the way it has been represented in the aforementioned website: whether it is a standard practice or not.
This is known as the shoelace formula.