DFT: Given input signals, is there a way to compute the real part of DFT output without writing out the summation?
The computation doesn't look correct. You have four 45 degree terms, the first two look OK, the second two do not. I'm no good keeping track of minus signs either! But apart from those angles, you're on the right route.

The summation is the most straightforward way to do it.

You could factor the summation, which is what the FFT does, to reduce the number of non-trivial multiplies. If you're trying to check for correctness, then that's not the way to go. If you're trying to speed it up, then there's little benefit in factoring for something as small as length 8, and no benefit if you only want a few m values.

An interesting alternative is the Geortzel algorithm, which is sometimes used for computing individual terms from a DFT. Understand the summation method before you tackle this.