Try to determine what it is about the scene or subject that interests you. In this case was it the lines and symmetry of cables. The implied strength of support column. Once you have that then frame the image to accentuate it. In your instance, I think you might have been trying to emphasis the lines of the cables. If so, then I would have tried a vertical orientation with the tower a bit more centered.
Another thing is to minimize distracting elements. The blue crane structures detract from it the bridge here. A vertical orientation would have allowed you to place those outside of your frame.
Point of view is another important element. Can you get closer to or farther away from your subject to accentuate your subject? Granted it can be tough for something as large as a bridge. However, it is possible.
A foreground object can also serve to direct the viewer's eye into the scene, provide a sense of action to the scene, or provide a compliment/opposite to the background. A boat or ship traveling under the bridge, or between you and the bridge might have helped as well.
Finally, time of day can dramatically change the subject. This particular bridge is lit up at night quite beautifully. Photographing it at night allows a photographer to include it's reflection in the water to the composition which is particularly helpful at emphasizing symmetry.
I hope that helps. I was down in Charleston a couple of months ago with my son's Cub Scout group. We spent the night on the USS Yorktown and thus I got the opportunity to photograph the bridge from a couple of angles and times of day that aren't generally available.
To illustrate my points, here are a couple of my shots of the bridge which I hope will help you.
This image was taken from the observation deck of the South Carolina Aquarium. It appears to be a bit further away from your point of view unless you were using a long zoom lens? My interest here was in showing the grace of the bridge across the water. That meant for me shooting a wide angle shot of the entire bridge. I framed the shot with more sky for a couple of reasons. One was to minimize the foreground elements that would detract from the bridge, i.e. railings, people, etc. that didn't add anything to the composition. Another was to give an impression of the bridge holding itself high above the water. When I processed it, I increased the saturation of the midtones to bring out the blue in the sky and water in order to provide a bit of color symmetry if you will which I think accents the bridge's symmetry. Also, notice how my distance from the bridge and the wide angle has minimized the same crane structure that was in your shot even though the complete structure is visible and dead center in mine. The point of view and wide angle makes it look much smaller and less distinct, thus removing the distraction and rendering just part of the background.
This image was made from the pier leading to teh USS Yorktown at twilight. Both the bridge and the Yorktown are lit at night and I wanted a photograph of the two of them together. Of course I couldn't move either of them, so I had to walk around to find an angle that I liked. It is framed so that the Yorktown and the finger of land to the right lead the eyes toward the bridge. The reflections in the water are for the over sense of symmetry. It was shot with a tungsten white balance to bring out the blue in the sky while registering the lights closer to white. I do wish I had found a view point that included the full reflection of the second tower in the water. However, my son was with me at the time and I had to pay attention to him as well which is pretty typical of my photo-outings ... I rarely get time to myself to just wander and explore a scene.
Again, I hope these helped. Keep practicing. You have an infinite amount of film available to you with digital. Thus, there is no cost to make mistakes and the only way to learn is to make lots of mistakes.