How to properly represent a sum of interactions in ANOVA?

I would like to include an ANOVA-style linear model in my work that shows the main effects but has interactions lumped together in one term for brevity. When I do the analysis I will include the proper interaction terms (well SAS will, but you get the gist).

The model has three factors: $$\alpha_i$$, $$\beta_j$$, and $$\gamma_k$$, each with three levels. The full model could be given by:

$$y_{ijkl} = \alpha_i + \beta_j + \gamma_k + \alpha\beta_{ij} + \alpha\gamma_{ik} + \beta\gamma_{jk} + \alpha\beta\gamma_{ijk} + error_{ijkl}$$

Where $$y$$ is the outcome variable, subscript $$i$$ refers to one of the three levels of factor $$\alpha$$, subscript $$j$$ refers to one of the three levels of factor $$\beta$$, subscript $$k$$ refers to one of the three levels of factor $$\gamma$$, and subscript $$l$$ refers to one of the replicates of a measurement made at a given combination of the other factor levels.

What I want to do is write it down the model like this:

$$y_{ijkl} = \alpha_i + \beta_j + \gamma_k + \omega_{ijk} + error_{ijkl}$$

Where $$\omega_{ijk}$$ refers to the sum of all the interaction terms listed in the full model. My question to someone with knowledge of notation is this: is the previous line preferable, or would it be better to list no subscripts on the proposed "lumped" or "stand-in" term? Is there any other way to elegantly do this? In reality my ANOVA has more than three factors, as you might have guessed, but I would like a succinct equation to represent it.