Anyone good with index notation and vector calculus?

Aha! I figured what I did wrong!

Stereo, δ[kk] isn't 1, it's the trace of I. Since this is in 3 dimensions, it's 1 + 1 + 1 = 3.

| δ[ir] δ[is] δ[ik] |
| δ[jr] δ[js] δ[jk] |
| δ[kr] δ[ks] δ[kk] |

+δ[ir].
| δ[js] δ[jk] |
| δ[ks] 3.... |
- δ[is].
| δ[jr] δ[jk] |
| δ[kr] 3.... |
+ δ[ik].
| δ[jr] δ[js] |
| δ[kr] δ[ks] |


3.δ[ir].δ[js] - δ[ir].δ[jk].δ[ks]
- 3.δ[is].δ[jr] + δ[is].δ[jk].δ[kr]
+ δ[ik].δ[jr].δ[ks] - δ[ik].δ[js].δ[kr]

= δ[ir]δ[js] - δ[is]δ[jr]

As required.