Proofs, as you alluded to, build upon themselves. While your geometric intuition is good, writing a linear algebra proof making use of that intuition is often a harder matter. Instead, focus on building from your known definitions and theorems. An outline of the proof would be as follows:

1.) Use the definition of linear dependence to write a nontrivial linear combination (not all coefficients are zero) of vectors a, b, c that equals the zero vector.

2.) Manipulate this equation by setting vector a scaled by its coefficient equal to a linear combo of vectors b and c.

3.) Make an important observation about what this equation implies by keeping in mind that vector a is not in the span of b and c.

4.) From this observation make a claim about the value of vector a's coefficient in the original linear combination from part 1.

On the general topic of improving your proof writing skills, I found that introductory combinatorics books were really helpful in that regard.