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I'm trying to solve $n\times(n \times (n\times x))$, I know that this can be achieved by simply using the definition of the cross product and I understand that it should equal $- n \times x$ conceptually, however I was wondering if there was a simplier way?

So some additional information is that n is a unit vector

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    use $a\times b\perp a$ – 2017-01-01
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    @Max can you elaborate I'm not sure what ⊥ is – 2017-01-01
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    it means perpendicular. the vector $a\times b$ is perpendiculat to $a$. EDIT: and if $c \perp d$ you know something: $\left\| c\times d\right\|=\left\| c\right\|\cdot \left\|d\right\|$ – 2017-01-01

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Expansion with the Grassmann Identity gives us

\begin{align} n\times (n\times(n\times x)) &= n\times(n(n\cdot x)-n^2x)\\ &= n\times(n\cdot x)n - n\times n^2x\\ &=n^2(n\times x) \end{align}