Let's imagine a one dimensional case, where a particle is moving with a velocity $v$ and an acceleration $a$. Thus
$$a=\frac{\mathrm dv}{\mathrm dt}\tag{1}$$
Applying the chain rule, equation $(1)$ can be rewritten as
$$a=\frac{\mathrm dv}{\mathrm dx}\frac{\mathrm dx}{\mathrm dt}\Longrightarrow \boxed{a=v\frac{\mathrm dv}{\mathrm dx}}\tag{2}$$
Now, if we were dealing with a 2D or a 3D case, then we would use vectors in the above expressions. Thus
\begin{alignat}{2} a&=\frac{\mathrm dv}{\mathrm dt}&&\Longrightarrow\mathbf a=\frac{\mathrm d \mathbf v}{\mathrm dt}\tag{3}\\ a&=v\frac{\mathrm dv}{\mathrm dx}&&\Longrightarrow \mathbf a=\:\:?\tag{4} \end{alignat}
As you can see, the vector form of equation $(1)$ (which is equation $(3)$) can be easily found, however I do not know of any way to express the equation $(2)$ in vector form.
The natural thought was to express the velocity into its components. For a 3D case, let $\mathbf v=v_x\mathbf{\hat i}+v_y\mathbf{\hat j}+v_z\mathbf{\hat k}$. Doing this, we have essentially converted the 3D case to three 1D cases. Thus using equation $(2)$:
$$\mathbf a =v_x\frac{\mathrm d v_x}{\mathrm dx}\mathbf{\hat i}+v_y\frac{\mathrm d v_y}{\mathrm dy}\mathbf{\hat j}+v_z\frac{\mathrm d v_z}{\mathrm dz}\mathbf{\hat k}\tag{5}$$
However, this expanded version doesn't seem particularly useful to me. Is there any way to express equation $(5)$ in a "closed form" (without explicitly writing out the components)? I feel that writing it in closed form might involve some common vector calculus operators (along with dot and cross products), though I am not exactly sure how to express it in a "closed form".
Justification of equation $(5)$: We kow that $\mathbf a=a_x\mathbf{\hat i}+a_y\mathbf{\hat j}+a_z\mathbf{\hat k}$
Now since
$$a_x=\frac{\mathrm dv_x}{\mathrm dt}=v_x\frac{\mathrm d v_x}{\mathrm d x}$$
Thus subsitituting this for every component, we re-obtain equation $(5)$.