I'm supposed to consider a Lorentz transformation of the form $\Lambda^{\mu}_{\ \nu} = \delta^{\mu}_{\ \nu} + \omega^{\mu}_{\ \nu}$, where $\omega$ is some tensor.

Since Lorentz tranformations satisfy $\Lambda^{\mu}_{\ \sigma} \eta^{\sigma \rho} \Lambda^{\nu}_{\ \rho}= \eta^{\mu \nu}$ (where $\eta$ is the Minkowski metric), I was able to find that $\omega$ satisfies the condition;

$\eta^{\mu \rho} \omega^{\nu}_{\ \rho} + \eta^{\nu \rho} \omega^{\mu}_{\ \rho} + \omega^{\mu}_{\ \sigma} \eta^{\sigma \rho} \omega^{\nu}_{\ \rho} = 0$

$\ $

Now my question is asking me to consider a Klein-Gordon field $\phi$, and Taylor expand it, so that I can find the variation $\delta \phi$ in terms of the $\omega$.

Where do I begin with this?

My attempt:

I'm assuming that I'm taking a tranformation $\phi(x) \mapsto \phi( x + \delta x ) = \phi(x) + \delta \phi$.

A guess I have, using a Taylor-type expansion, is:

$\phi( x + \delta x ) = \phi(x) + (\delta x)^{\mu} \partial_{\mu} \phi(x) + ....$, so that $\delta\phi \approx (\delta x)^{\mu} \partial_{\mu} \phi(x)$

But I am really quite lost from here...how do I incorporate $\omega$ into the above?

EDIT: I made a mistake; the tensor $\omega$ is infinitesimal, and this condition results in $\omega^{\mu \nu} = - \omega^{\nu \mu}$, so that $\omega$ is anti-symmetric.