= Solution
Contracting two antisymmetric tensors gives the nonzero isotropic rank-four tensor
$$
\boxed{
\epsilon_{ijm}\epsilon_{klm}
=\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}}.
$$
A nonzero isotropic rank-five tensor is, for example,
$$
\boxed{T_{ijklm}=\epsilon_{ijk}\delta_{lm}}.
$$
More generally, one obtains the rank-five isotropic building blocks by choosing which two of the five indices belong to the Kronecker delta and putting the remaining three into the Levi-Civita symbol. There are
$$
\binom52=10
$$
such placements:
$$
\epsilon_{ijk}\delta_{lm},
\epsilon_{ijl}\delta_{km},
\epsilon_{ijm}\delta_{kl},
\epsilon_{ikl}\delta_{jm},
\epsilon_{ikm}\delta_{jl},
\epsilon_{ilm}\delta_{jk},
$$
$$
\epsilon_{jkl}\delta_{im},
\epsilon_{jkm}\delta_{il},
\epsilon_{jlm}\delta_{ik},
\epsilon_{klm}\delta_{ij}.
$$
\b[Thus the unrestricted isotropic rank-five ansatz requires ten <scalar> component coefficients, one for each delta-epsilon placement.]
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