I'm going to assume that "Euler characteristic with compact support" means

"(Euler characteristic of the one point compactification) - 1".

Let me assume that n>1.

The space in question, namely $ GL ( n,R) _ +$ , has a circle action given by any $ S ^ 1 $ subgroup of $ GL(n,R) $. This action is free on $ GL(n,R) $, and fixes the point at infinity. $ S ^ 1 $-orbits contribute zero to the euler characteristic, and the point at infinity contributes 1.
So $ \chi ( GL (n,R) _ +) = 1 $, and the Euler characteristic with compact support is zero.

To make te above argument precise, you need to pick a cell decomposition of $ ( GL ( n,R)/S ^ 1 ) _ + $,
and use it to construct a cell decomposition of $GL ( n,R)$. Above every n-cell of the quotient space, you put a pair of cells of $GL ( n,R) _ + $, one of dimension n and one of dimension n+1 (except for the 0-cell corresponding to the point at infinity). This might fail to be a CW-complex, but you can nevertheless compute the Euler characteristic as the alternating sum of the numbers of cells in given dimensions.